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Hindawi Publishing CorporationInternational Journal of AnalysisVolume 2013 Article ID 878051 15 pageshttpdxdoiorg1011552013878051

Research ArticleMathematical Analysis and Numerical Simulations for a SystemModeling Acid-Mediated Tumor Cell Invasion

Christian Maumlrkl1 Guumllnihal Meral2 and Christina Surulescu3

1 Institut fur Numerische und Angewandte Mathematik Universitat Stuttgart Pfaffenwaldring 57 70569 Stuttgart Germany2Department of Mathematics Faculty of Arts and Sciences Bulent Ecevit University 67100 Zonguldak Turkey3 Technische Universitat Kaiserslautern Felix Klein Zentrum fur Mathematik Paul Ehrlich Strasse 67663 Kaiserslautern Germany

Correspondence should be addressed to Christina Surulescu surulescumathematikuni-klde

Received 20 September 2012 Revised 26 December 2012 Accepted 3 January 2013

Academic Editor Liancheng Wang

Copyright copy 2013 Christian Markl et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This work is concerned with the mathematical analysis of a model proposed by Gatenby and Gawlinski (1996) in order to supportthe hypothesis that tumor-induced alteration of microenvironmental pH may provide a simple but comprehensive mechanism toexplain cancer invasion We give an intuitive proof for the existence of a solution under general initial conditions upon using aniterative approach Numerical simulations are also performed which endorse the predictions of the model when compared withexperimentally observed qualitative facts

1 Introduction

Despite major progress in medicine and science there still areincurable diseases which can threaten human lives Canceris among the most severe ones and it manifests itself asan uncontrolled growth of cells which are produced by theorganism subsequently to mutations Cancer cells migratethrough the surrounding tissue and degrade it on their waytoward blood vessels and distal organs where they initiate anddevelop further tumors a process known as metastasis

In the last decades various classes of models have beenproposed aiming to provide a quantitative description oftumor growth They range from the microscopic level ofintracellular signaling pathways conditioning the growth ofneoplastic tissue by stimulation or inhibition of apoptosis(eg by the influence of tumor necrosis factors [1]) or tumorcell motility for example by restructuring the cytoskeletonor by producing matrix degrading enzymes [2] through thelevel of cell-cell or cell-tissue interactions and up to themacroscopic level characterizing the behavior of the entirecell population Multiscale settings like those in [3 4] involveseveral of these scales and offer a systemic approach to themodeling process

When ignoring the setups relying on mechanical forcebalance andor on the theory of mixtures in the study of

tumor invasion and metastasis one can distinguish betweenthe so-called kinetic approach and the direct modeling atthe macroscopic level In the former a mesoscopic modelis considered consisting of an integro-partial differentialequation for the evolution of the cell density possibly coupledwith integro-differential andor reaction-diffusion equationsfor the fibre density of the extracellularmatrix (ECM) and thechemotactic signal (see eg [5] and the references therein[3 4]) Then with an appropriate scaling the macroscopiclimit is deduced usually leading to a Keller-Segel-type modelor some hyperbolic systems see for example [6] Themacroscopic approach involves the largest class of existingmodels and directly accounts for processes at the level ofcell populations leading to systems of reaction-diffusion(transport) equations like for example in [7 8]

The role of tumormicroenvironment in determining can-cermalignancy has been put in evidence in several referencessee for example [9 10] For instance hypoxia and acidityare factors that can trigger the progression from benign tomalignant growth In order to survive in the unfavourableenvironment they create cancer cells upregulate certainproton extrusion mechanisms [11] the consequence of whichis that the extracellular tumor environment has an acidic pHwhich boosts apoptosis of normal cells and thus allows theneoplastic tissue to extend in the space becoming available

2 International Journal of Analysis

Hence the pH level directly influences the metastatic poten-tial of tumor cells [12 13] These facts led Gatenby andGawlinski [8 14] to propose a model for the acid-mediatedtumor invasion which describes the interaction between thedensity of normal cells tumor cells and the concentrationof H+ protons produced by the latter via reaction-diffusionequations Starting from this model travelling waves havebeen used to explain the aggressive action of cancer cells ontheir surroundings [15] Further settings issued fromGatenbyand Gawlinskirsquos model involve nutrient dynamics influencedby both vascular and avascular growth of multicellular tumorspheroids [16 17] assuming rotational symmetry and investi-gating existence and qualitative properties of the solutions

In this work we reconsider the model in [8] whereby alsoexplicitly allowing for crowding effects (due to competitionwith cancer cells) in the growth of normal cells (Thiscan be also done for the growth term in the tumor cellequation however it does not change the analysis nor thequalitative behavior of the solutions to the system Moreoverits biological motivation is not strong enough since thecompetition between the two cell types does not reallyaffect the growth but rather the invasion of the neoplastictissue the acidity increase in the peritumoral environmentis a byproduct of the enhanced glycolysis of cancer cellsand not produced with the purpose of killing the normalcells and eluding concurrence) For this setting we performmathematical analysis and numerical simulations in order toverify the model predictions with respect to experimentallyobserved qualitative facts In order to prove the existence ofa unique (weak) solution we propose an intuitive methodrelying on an iterative procedure which has also been appliedin [18 19] in a different context (one of the supplementarydifficulties here is the diffusion coefficient being nonconstantbut depending on the solution itself) and allows to avoid theuse of operator semigroups

2 Problem Setting

The model by Gatenby and Gawlinski [8] describes theevolution of normal and tumor cell density respectively ina domain where these cell types interact on the basis of pHvalue modifications The mathematical description of theseprocesses is ensured by the following system of reaction-diffusion equations for the normal cell density 119873(119905 x) thetumor cell density 119870(119905 x) (both in cellscm3) and theconcentration of excessive H+(119905 x) ion concentration (inMol)

120597119873

120597119905= 119908119873119873(1 minus

119873

119870119873

minus 120579119870

119870119870

) minus 119889119873119867119873 in (0 119879) times Ω

120597119870

120597119905= 119908119870119870(1 minus

119870

119870119870

) + nabla sdot (119863119870(1 minus

119873

119870119873

)nabla119870)

in (0 119879) times Ω

120597119867

120597119905= 119908119867119870 minus 119889

119867119867 + 119863

119867Δ119867 in (0 119879) times Ω

(1)

where 120579 le 12 denotes the strength parameter for thecompetition between normal cells and neoplastic tissueThereby Ω sub R119899 (119899 = 1 2 3) is a regular-enough andbounded domain and only microscopically small processesare considered at the interface between tumor and healthytissue Observe that the diffusion coefficient of the cancercells depends on the normal cell density when the healthytissue is at its carrying capacity the neoplastic tissue cannotdiffuse thus the tumor is confined It can only spread if thesurrounding normal tissue is diminished from its carryingcapacity and this is assumed to happen due to lowering thepH level upon secretion of H+ protons by cancer cells

The constants 119908119873and 119908

119870are given in 1s and represent

the growth rates and the constants119870119873and119870

119870are expressed

in cellscm3 and provide the carrying capacities of the normaland tumor cells respectivelyThe death rate 119889

119873of the normal

cells is measured in 1(Mol sdot s) and the diffusion coefficientof tumor cells in the absence of normal cells is given incm2s The production rate 119908

119867of H+ protons is expressed

inMol sdot cm3(cells sdot s) the uptake rate 119889119867(due for instance to

proton buffering (see eg [20] and the references therein)andor various ion exchangers between intracellular andextracellular domains (see eg [21])) is measured in 1s andtheir diffusion coefficient of119863

119867in cm2s

We also assume that there is no exchange of cells andH+ protons through the boundary of the considered domainthus

120597119870

120597n=120597119867

120597n= 0 in (0 119879) times 120597Ω (2)

where n denotes the outer unit normal vector to 120597ΩThe initial conditions are given by

119870 (0 x) = 1198700(x) 119873 (0 x) = 119873

0(x)

119867 (0 x) = 1198670(x) in Ω

(3)

Thereby 1198700(x) 119873

0(x) and 119867

0(x) are strictly positive func-

tions which satisfy the no-flux condition

1205971198700

120597n=1205971198670

120597n= 0 on 120597Ω (4)

In order to render the system (1) dimensionless we use thefollowing transformations

=119873

119870119873

=119870

119870119870

= 119867 sdot119889119867

119908119867119870119870

= 119908119873sdot 119905 x = radic

119908119873

119863119867

sdot x(5)

along with the notations

120575119873=119889119873119908119867119870119870

119889119867119908119873

120588119870=119908119870

119908119873

Δ119870=119863119870

119863119867

120575119867=119889119867

119908119873

(6)

International Journal of Analysis 3

We obtain the system

120597119873

120597119905= 119873 (1 minus 119873 minus 120579119870) minus 120575

119873119867119873

120597119870

120597119905= 120588119870119870 (1 minus 119870) + nabla sdot (Δ

119870(1 minus 119873)nabla119870)

120597119867

120597119905= 120575119867119870 minus 120575

119867119867 + Δ119867

(7)

where for simplicity the tilde notations have been ignoredThe stability analysis of this system (with 120579 = 0) hasbeen performed in [8] leading to biologically significantpredictions

3 Existence and Uniqueness of Solutions

In this section we provide a natural proof for the existenceand uniqueness of a weak solution to the system (1) withinitial data (3) and boundary conditions (2) We make useof an iterative procedure instead of the classical approachvia semigroup theory this is more intuitive and allows for aseparate treatment of the three equations in each step

Consider the function spaces

119883 = 119871infin

(0 1198791198671

(Ω))

119884 = 119906 isin 1198712

(0 1198791198672

(Ω)) 119906119905isin 1198712

(0 119879 1198712

(Ω))

119885 = 119871infin

(0 119879 1198712

(Ω))

(8)

Definition 1 Aweak solution of (1) with boundary conditions(2) and initial data (3) is a triple (119867119873119870) of functions in119883times119884 times 119885 such that for all 120601 isin 1198671(Ω) ae in [0 119879] the followingthree equations are satisfied

intΩ

119908119867119870120601119889x

= intΩ

119867119905120601119889x + int

Ω

119863119867nabla119867nabla120601119889x + int

Ω

119889119867119867120601119889x

intΩ

119908119873119873(1 minus

119873

119870119873

minus 120579119870

119870119870

)120601119889x

= intΩ

119873119905120601119889x + int

Ω

119873119889119873119867120601119889x

intΩ

119908119870119870(1 minus

119870

119870119870

)120601119889x

= intΩ

119870119905120601119889x + int

Ω

119863119870(1 minus

119873

119870119873

)nabla119870nabla120601119889x

(9)

Theorem 2 There exists 119879 gt 0 such that the system (1) withinitial data (3) and boundary conditions (2) satisfying

1198670isin 1198671

(Ω) cap 119862 (Ω) 1198730isin 119871infin

(Ω) cap 1198671

(Ω)

1198700isin 1198671

(Ω)

1198670ge 119862119867gt 0 0 lt 119873

0le119870119873

2 0 lt 119870

0le 119870119870

(10)

has a unique solution (119867119870) isin (119883 times119883) cap (119884 times119884) and119873 isin 119885

We set

119879 =

6

prod

119894=1

119879119894

(11)

with 119879119894le 1 to be defined below

In order to proveTheorem 2 we construct a sequence

(119867119898

119870119898

)119898isinN0

isin (119883 times 119883) cap (119884 times 119884)

(119873119898

)119898isinN0

isin 119885

(12)

and prove its convergence towards the weak solution of thesystem

Let (1198670 1198700) isin (119883times119883)cap (119884times119884) and1198730 isin 119885 be the weaksolution to the homogeneous system

1205971198670

120597119905minus 119863119867Δ1198670

+ 1198891198671198670

= 0 (13)

1205971198730

120597119905+ 1198891198731198670

1198730

= 0 (14)

1205971198700

120597119905minus nabla sdot (119863

119870(1 minus

1198730

119870119873

)nabla1198700

) = 0 (15)

while (119867119898 119870119898)119898isinN0

isin (119883 times 119883) cap (119884 times 119884) and (119873119898)119898isinN0

isin 119885

is the weak solution to

120597119867119898+1

120597119905minus 119863119867Δ119867119898+1

+ 119889119867119867119898+1

= 119908119867119870119898

(16)

120597119873119898+1

120597119905+ 119889119873119867119898+1

119873119898+1

= 119908119873119873119898

(1 minus119873119898

119870119873

minus 120579119870119898

119870119870

)

(17)

120597119870119898+1

120597119905minus nabla sdot (119863

119870(1 minus

119873119898+1

119870119873

)nabla119870119898+1

)

= 119908119870119870119898

(1 minus119870119898

119870119870

)

(18)

with the corresponding initial and boundary conditions (2)and (3)

The existence and uniqueness of the functions (119867119898119870119898 119873119898)119898isinN0

in the above sequence are ensured by thefollowing

Lemma 3 (properties of the iteration sequence) Underassumptions (10) there exists 119879 gt 0 such that

(i) there exists a unique weak solution to the systems (13)ndash(15) and (16)ndash(18) with conditions (3) and (2) and forevery119898 isin N

0it holds that

119873119898

119873119898

119905isin 119871infin

((0 119879] times Ω) (19)

119867119898

119870119898

isin 1198712

(0 1198791198672

(Ω)) cap 119871infin

(0 1198791198671

(Ω))

119867119898

119905 119870119898

119905isin 1198712

(0 119879 1198712

(Ω))

(20)


(ii) the functions119867119898119873119898 and119870119898 are positive for all119898 isin

N0 Moreover the following inequalities hold

119867119898

(119905 x) ge 119862119867119890minus119889119867119905 119873

119898

(119905 x) le 119870119873

2

119870119898

(119905 x) le 119870119870

for ae x isin Ω 119905 isin [0 119879] (21)

(iii) the functions 119867119898 119873119898 and 119870119898 satisfy for adequateconstants 119862(Ω 119879) and for all119898 isin N

0the estimates

10038171003817100381710038171198671198981003817100381710038171003817119883

+100381710038171003817100381711986711989810038171003817100381710038171198712(01198791198672(Ω))

le 119862 (Ω 119879) (10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)+10038171003817100381710038171198670

10038171003817100381710038171198671(Ω))

(22)

10038171003817100381710038171198731198981003817100381710038171003817

2

119883le 119862 (Ω 119879)

100381710038171003817100381711987301003817100381710038171003817

2

1198671(Ω) (23)

10038171003817100381710038171198701198981003817100381710038171003817119883

+100381710038171003817100381711987011989810038171003817100381710038171198712(01198791198672(Ω))

le 2119862 (Ω 119879)10038171003817100381710038171198700

10038171003817100381710038171198671(Ω) (24)

Remark 4 From (19) it follows that

119873119898

isin 119871infin

(0 119879 1198712

(Ω)) (25)

for all119898 isin N0

Proof of Lemma 3 We performmathematical induction withrespect to119898

Induction StartThe proof of the claims in Lemma 3 for119898 = 0

is done separately for each of (13)ndash(15)(a) With the substitution

0

(119905 x) = 1198670 (119905 x) 119890119889119867119905 (26)

Equation (13) becomes the heat equation

0

119905minus 119863119867Δ0

= 0 (27)

thus by the theory of linear parabolic differential equations(see eg [22]) and with the assumption 119867

0isin 1198671(Ω) it

follows that there exists a unique solution1198670 of (13) such that

1198670

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

1198670

119905isin 1198712

(0 119879 1198712

(Ω))

(28)

This weak solution also satisfies10038171003817100381710038171003817119867010038171003817100381710038171003817119883

+100381710038171003817100381710038171198670100381710038171003817100381710038171198712(01198791198672(Ω))

le 119862 (Ω 119879)10038171003817100381710038171198670

10038171003817100381710038171198671(Ω) (29)

Further it is known (see eg [23]) that the solution of (13)can be written explicitly with respect to the initial condition1198670and the heat kernel and it is therefore positive(b) Equation (14) is linear and has a positive solution

1198730

(119905 x) = 1198730119890minusint

119905

01198891198731198670(119904x)119889119904

gt 0 (30)

which depends on1198670(119905 x) It follows immediately that

10038171003817100381710038171003817119873010038171003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

=

10038171003817100381710038171003817100381710038171198730119890minusint

119905

01198891198731198670119889119905

1003817100381710038171003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

le10038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

(31)

and thus the estimation (23) for119898 = 0 is obtainedThe corresponding statement (19) for1198730 is to be justified

below(c) In order to prove the claims of Lemma 3 for 1198700 we

show first that

1198730

isin 119871infin

((0 119879] times Ω) (32)

1198730


((0 119879] times Ω) (33)

The former follows from10038171003817100381710038171003817119873010038171003817100381710038171003817119871infin((0119879]timesΩ)

(30)

=

10038171003817100381710038171003817100381710038171198730119890minusint

119905

01198891198731198670119889119905

1003817100381710038171003817100381710038171003817119871infin((0119879]timesΩ)

le10038171003817100381710038171198730

1003817100381710038171003817119871infin(Ω)lt infin

(34)

For 119905 ge 120575 gt 0 it is100381710038171003817100381710038171198730

119905

10038171003817100381710038171003817119871infin((0119879]timesΩ)

(30)

= 119889119873

10038171003817100381710038171003817100381710038171198730sdot 119890minusint

119905

01198891198731198670119889119905

sdot 1198670

1003817100381710038171003817100381710038171003817119871infin((0119879]timesΩ)lt infin

(35)

For 119905 rarr 0we can consider (27) For its solution it holds (seeeg [22]) that

lim(119905x)rarr (0x0)

0

(119905 x) = 1198670(x0) for every x0 isin Ω (36)

Therefore


1198670

(119905 x) = lim(119905x)rarr (0x0)

0

(119905 x) 119890minus119889119867119905 = 1198670(x0)

(37)

and finally (33) follows thus also (19) for119898 = 0The following proof of (20) and (24) upon starting from

(15) relies on Theorem 715 in Evans [22] However thatresult cannot be directly applied to the present case since thediffusion coefficient 119886(119905 x) = 119863

119870(1 minus (119873

0(119905 x)119870

119873)) in (15)

depends on timeLet

119896119898(119905) =

119898

sum

119894=1

119889119894

119898(119905) 119908119894

(38)

with functions 119908119894= 119908119894(x) such that

119908119894infin

119894=1is an orthogonal basis of 1198671 (Ω)

119908119894infin

119894=1is an orthonormal basis of 1198712 (Ω)

(39)

Considering the symmetric bilinear form

119860 [119896119898 119896119898] = intΩ

119886 (119905 x) (nabla119896119898)2

119889x (40)


the dependence of the coefficient 119886(119905 x) on 119905 leads in its timederivative

119889

119889119905119860 [119896119898 119896119898] = intΩ

1198861015840

(119905 x) (nabla119896119898)2

119889x

+ 2intΩ

119886 (119905 x) (nabla119896119898)1015840

nabla119896119898119889x

(41)

to a supplementary summand

intΩ

1198861015840

(119905 x) (nabla119896119898)2

119889x = minusintΩ

119863119870

119870119873

(1198730

)1015840

(119905 x) (nabla119896119898)2

119889x

(42)

where for shortness we denoted by 1015840 the derivative withrespect to 119905

The rest of the proof ofTheorem 715 in [22] can now beadapted to obtain for an arbitrary 120577 gt 0 the estimate

100381710038171003817100381710038171198961015840

119898

10038171003817100381710038171003817

2

1198712(Ω)

+119889

119889119905(1

2119860 [119896119898 119896119898])

le119862

120577(1003817100381710038171003817119896119898

1003817100381710038171003817

2

1198671(Ω)

+10038171003817100381710038171198911003817100381710038171003817

2

1198712(Ω))

+ 2120577100381710038171003817100381710038171198961015840

119898

10038171003817100381710038171003817

2

1198712(Ω)

+1

2intΩ

119863119870

119870119873

(1198730

)1015840

(nabla119896119898)2

119889x

(43)

Now let (recall (33))

1198721198730 =

119863119870

119870119873

100381710038171003817100381710038171198730

119905

10038171003817100381710038171003817119871infin((0119879]timesΩ) (44)

Upon integrating with respect to 119905 one can majorize

int

119879

0

intΩ

119863119870

119870119873

(1198730

)1015840

(nabla119896119898)2

119889x 119889119905 le 1198721198730 int

119879

0

1003817100381710038171003817nabla1198961198981003817100381710038171003817

2

1198712(Ω)119889119905

le 1198721198730

10038171003817100381710038171198961198981003817100381710038171003817

2

1198712(0119879119867

1(Ω))

le 120574 (Ω 119879) lt infin

(45)

with 120574(Ω 119879) an adequate constant The rest of the proof canbe done as inTheorem 715 in [22] upon taking into account(32) and 119870

0isin 1198671(Ω) in order to show that there exists a

unique weak solution1198700(119905 x) to (15) such that

1198700

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

1198700

119905isin 1198712

(0 119879 1198712

(Ω))

10038171003817100381710038171003817119870010038171003817100381710038171003817119883

+100381710038171003817100381710038171198700100381710038171003817100381710038171198712(01198791198672(Ω))

le 119862 (Ω 119879)10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)

(46)

Since

1198700

0(x) gt 0 (47)

it follows from the weakmaximumprinciple that1198700(119905 x) gt 0and thus also the positivity of1198700(119905 x)

The proof of the inequalities (21) for119898 = 0 does not differfrom the one for a general119898 isin N given below and is thereforeomitted here

With (a)ndash(c) we proved all statements of Lemma 3 for119898 = 0

Induction Hypothesis Assume the assertions of the lemmahold for an arbitrary119898 isin N

0

Inductive Step The proof for 119898 + 1 is to be done separatelyfor each of (16)ndash(18) Since for a corresponding embeddingconstant 119888

1= 1198881(Ω 119879)

int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198712(Ω)119889119905 le 119888

1int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)119889119905

ind hyple(24)

411988811198622

(Ω 119879) 11987910038171003817100381710038171198700

1003817100381710038171003817

2

1198671(Ω)

lt infin

(48)

and thus

119870119898

isin 1198712

(0 119879 1198712

(Ω)) (49)

the existence of a unique weak solution to (16) (2) and(3) follows from the theory of linear parabolic differentialequations The solution119867119898+1(119905 x) satisfies

119867119898+1

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

119867119898+1

119905isin 1198712

(0 119879 1198712

(Ω))

10038171003817100381710038171003817119867119898+110038171003817100381710038171003817119883

+10038171003817100381710038171003817119867119898+1100381710038171003817100381710038171198712(01198791198672(Ω))

le 1198621(Ω 119879) (2119908

119867119862 (Ω 119879)radic119888

111987910038171003817100381710038171198700

10038171003817100381710038171198671(Ω)+10038171003817100381710038171198670

10038171003817100381710038171198671(Ω))

le C (Ω 119879) (10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)+10038171003817100381710038171198670

10038171003817100381710038171198671(Ω))

(50)

withC(Ω 119879) = max1198621(Ω 119879) 119862

1(Ω 119879)2119908

119867119862(Ω 119879)radic119888

1119879

In order to establish the lower bound for119867119898+1 define anauxiliary function 120595119898+1(119905 x) = 119867

119898+1(119905 x) minus 119862

119867119890minus119889119867119905 for

which it holds

⟨120595119898+1

119905(119905) 120601⟩ + 119863

119867intΩ

nabla120595119898+1

nabla120601119889x + 119889119867intΩ

120595119898+1

120601119889x

= ⟨119908119867119870119898

120601⟩ (51)

For every nonnegative 120601 isin 1198671(Ω) the right-hand side is

positive Further 120595119898+1(0 x) ge 0 by construction thus itfollows with the weak maximum principle that 120595119898+1 ge 0 aewhich leads to119867119898+1(119905 x) ge 119862

119867119890minus119889119867119905

Now (17) is a linear inhomogeneous differential equationwith solution

119873119898+1

(119905 x) = 119890minus120572(119905x) (1198730(x) + int

119905

0

120573 (119904 x) 119890120572(119904x)119889119904) (52)


where

120572 (119905 x) = int119905

0

119889119873119867119898+1

(V x) 119889V

120573 (119904 x) = 119908119873119873119898

(119904 x) (1 minus 119873119898

(119904 x)119870119873

minus 120579119870119898

(119904 x)119870119870

)

(53)

In order to prove (19) for119898 + 1 we have to show that

119873119898+1

isin 119871infin

((0 119879] times Ω) (54)

119873119898+1


((0 119879] times Ω) (55)

Obviously the first assertion (54) holds due to the inductionhypothesis

Next estimate10038171003817100381710038171003817119873119898+1

119905

10038171003817100381710038171003817119871infin((0119879]timesΩ)

le 119908119873

10038171003817100381710038171198731198981003817100381710038171003817119871infin((0119879]timesΩ)

10038171003817100381710038171003817100381710038171003817

1 minus119873119898

119870119873

minus 120579119870119898

119870119870

10038171003817100381710038171003817100381710038171003817119871infin((0119879]timesΩ)

+ 119889119873

10038171003817100381710038171003817119873119898+110038171003817100381710038171003817119871infin((0119879]timesΩ)

10038171003817100381710038171003817119867119898+110038171003817100381710038171003817119871infin((0119879]timesΩ)

lt infin

(56)

due to (19)Using again the induction hypothesis the regularity of the

initial data and the properties of the solutions to the heatequations it follows immediately that 119867119898+1

119871infin((0119879]timesΩ)

lt infinwhich leads to

10038171003817100381710038171003817119873119898+1

119905

10038171003817100381710038171003817119871infin((0119879]timesΩ)lt infin (57)

and thus (55) is provedNow we prove the positivity of 119873119898+1 and the corre-

sponding inequality in (21) To this aim use the inductionhypothesis to observe that

119873119898+1

(119905 x) le 119870119873

2119890minus120572(119905x)

+ int

119905

0

120573 (119904 x) 119890minus(120572(119905x)minus120572(119904x))119889119904

le119870119873

2119890minus120572(119905x)

+ 119908119873

119870119873

2int

119905

0

119890minus(120572(119905x)minus120572(119904x))

119889119904

(58)

Next notice that there exists a positive constant 119867such that

119867119898+1

(119905 x) ge 119867for ae x isin Ω 119905 isin [0 119879] This leads to the

estimate

119873119898+1

(119905 x) le 119870119873

2119890minus119889119873119867119905 + 119908

119873

119870119873

2

1

119889119873119867

(1 minus 119890minus119889119873119867119905)

le119870119873

2((1 minus

119908119873

119889119873119867

) 119890minus119889119873119867119905 +

119908119873

119889119873119867

) le119870119873

2

(59)

This in turn immediately implies via (52) the positivity of119873119898+1In the next step we prove the estimate (23) for119873119898+1(119905 x)

Due to (52) we get

10038171003817100381710038171003817119873119898+1

(119905)10038171003817100381710038171003817

2

1198671(Ω)

=

10038171003817100381710038171003817100381710038171003817

119890minus120572(119905)

1198730+ 119890minus120572(119905)

int

119905

0

120573(119904)119890120572(119904)

119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 21199082

119873

10038171003817100381710038171003817100381710038171003817

int

119905

0

119873119898

(119904) (1 minus119873119898(119904)

119870119873

minus 120579119870119898(119904)

119870119870

)119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 21199082

119873

100381710038171003817100381710038171003817100381710038171003817

int

119905

0

(119873119898

(119904) minus(119873119898(119904))2

1198702

119873

)119889119904

100381710038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 41199082

119873(

10038171003817100381710038171003817100381710038171003817

int

119905

0

119873119898

(119904)119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

+1

1198702

119873

10038171003817100381710038171003817100381710038171003817

int

119905

0

(119873119898

(119904))2

119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

)

le10038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)[2 + 4119908

2

119873119862 (Ω 119879) 119879

2

]

le C (Ω 119879)10038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

(60)

by (23) and the induction hypothesisIn order to prove the assertions of Lemma 3 for119870119898+1(119905 x)

one can apply Theorem 715 in [22] with (54) (55) and thesame justification as for the induction start at (c)

With an adequate embedding constant 1198882= 1198882(Ω 119879)

int

119879

0

10038171003817100381710038171003817100381710038171003817

119870119898

(1 minus119870119898

119870119870

)

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le int

119879

0

(100381710038171003817100381711987011989810038171003817100381710038171198712(Ω)

+

1003817100381710038171003817100381710038171003817100381710038171003817

(119870119898)2

119870119870

10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

)

2

119889119905

le 2int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198712(Ω)119889119905 + 2int

119879

0

1003817100381710038171003817100381710038171003817100381710038171003817

(119870119898)2

119870119870

1003817100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 21198882

1int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)119889119905 + 2

1198884

2

1198702

119870

int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

4

1198671(Ω)119889119905

le 81198882

11198622

(Ω 119879)10038171003817100381710038171198700

1003817100381710038171003817

2

1198671(Ω)11987911198792

+ 321198884

2

1198702

119870

1198624

(Ω 119879)10038171003817100381710038171198700

1003817100381710038171003817

4

1198671(Ω)11987911198792lt infin

(61)

by (24) and the induction hypothesis therefore 119870119898(1 minus(119870119898119870119870)) isin 119871

2(0 119879 119871

2(Ω)) and119870

0(x) isin 1198671(Ω) By applying


Theorem 715 in [22] it follows that (18) has a unique weaksolution119870119898+1(119905 x) with

119870119898+1

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

119870119898+1

119905isin 1198712

(0 119879 1198712

(Ω))

(62)

Now choose 1198791such that max119879

11198622(Ω 119879) 119879

11198624(Ω 119879) le 1

and

1198792= min1

2

1

161199082

1198701198882

1

100381710038171003817100381711987001003817100381710038171003817

1198702

119870

641199082

1198701198884

2

100381710038171003817100381711987001003817100381710038171003817

3 (63)

Then

int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905 le10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)(64)

and thus the estimate10038171003817100381710038171003817119870119898+110038171003817100381710038171003817119883

+10038171003817100381710038171003817119870119898+1100381710038171003817100381710038171198712(01198791198672(Ω))

le 2119862 (Ω 119879)10038171003817100381710038171198700

10038171003817100381710038171198671(Ω) (65)

holdsIn order to prove the positivity of 119870119898+1 we introduce an

auxiliary function

120585119898+1

(119905 x) = minus119860119905 exp (1 minus 120578119905) + 119870119870minus 119870119898+1

(119905 x) (66)

for 119860 positive and large enough and 120578 a positive constant tobe correspondingly chosen (see below) With the aid of thisfunction we show that for all119898 isin N

0

119870119898+1

le 119870119870 (67)

on an adequate time interval

Proof (of the Statement (67))

Induction StartThe proof of (67) for119898 = 0 is identical to theone for119898 + 1

Induction Hypothesis Assume assertion (67) holds for anarbitrary119898 isin N

0

Inductive Step Upon using (66) in (18) we get

120597120585119898+1

120597119905minus nabla sdot (119863

119870(1 minus

119873119898+1

119870119873

)nabla120585119898+1

)

= 119860 (120578119905 minus 1) exp (1 minus 120578119905) minus 119908119870119870119898

(1 minus119870119898

119870119870

)

(68)

Since

119870119898

(1 minus119870119898

119870119870

) le 119870119870 (69)

for the right-hand side of (68) we have that

119860 (120578119905 minus 1) exp (1 minus 120578119905) minus 119908119870119870119898

(1 minus119870119898

119870119870

) ge 0 (70)

holds for 119905 lt 1198793with correspondingly chosen 119879

3and 120578 such

that 120578119905 gt 1Since by construction 120585119898+1(0 x) ge 0 we can apply the

weak maximum principle for 119905 le 1198793to show that

119860 (120578119905 minus 1) exp (1 minus 120578119905) + 119870119870minus 119870119898+1

(119905 x) = 120585119898+1 ge 0 (71)

from which it also follows that

119870119898+1

le 119870119870 (72)

This completes the proof of the statement (67)In virtue of (67) for 119879 le 1119908

119870the right-hand side in (18)

is positive Since by hypothesis119870119898+10

gt 0 the weakmaximumprinciple implies the positivity of 119870119898+1 This ends the proofof all statements in Lemma 3 for an arbitrary 119898 isin N

0and

therefore the proof of the lemma itself

Now we are able to pass to the following

Proof (of Theorem 2)Existence In order to prove the existence of a weak solu-tion to (1) and (2) we show that the iterative sequence(119873119898 119870119898 119867119898)119898isinN0

is CauchyDue to the completeness of 1198671(Ω) and 119871

2(Ω) this

will imply the convergence of the sequence to some limitfunctions119873119870 and119867 these being solutions to (1) and (2)

Consider an arbitrary119898 isin N0 Since119867119898

0 119867119898+1

0isin 1198671(Ω)

and119870119898 119870119898+1 isin 1198712(0 119879 1198712(Ω)) it follows that

119867119898+1

0minus 119867119898

0isin 1198671

(Ω)

119870119898+1

minus 119870119898

isin 1198712

(0 119879 1198712

(Ω))

(73)

Next one can apply Theorem 715 in [22] to the difference119867119898+1

minus 119867119898 to deduce the estimate

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817119908119867119870119898

minus 119908119867119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

(74)

The right-hand side above can be further estimated and withthe embedding constant 119888

3= 1198883(Ω 119879) it follows that

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879)1199082

1198671198882

3int

119879

0

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le 119862 (Ω 119879)1199082

1198671198882

31198794

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

2

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(75)

where

1198794= min1

4

1

4119862 (Ω 119879)1199082

1198671198882

3

(76)


In order to obtain a corresponding estimate for the sequence(119873119898)119898isinN consider two consecutive terms in (17) written for

119873119898 and119873119898+1 and substract This leads to

120597

120597119905(119873119898+1

minus 119873119898

) + 119889119873(119867119898+1

119873119898+1

minus 119867119898

119873119898

)

= 119908119873(119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)

minus119873119898minus1

(1 minus119873119898minus1

119870119873


119870119870

))

(77)

Denote ℎ(119873119898 119873119898minus1) = 119908119873(119873119898(1minus(119873

119898119870119873)minus(119870

119898119870119870))minus

119873119898minus1

(1 minus (119873119898minus1

119870119873) minus (119870

119898minus1119870119870)))

Now multiply with (119873119898+1

minus 119873119898) and integrate with

respect to x to infer

1

2intΩ

120597

120597119905(119873119898+1

minus 119873119898

)2

119889x

+ 119889119873intΩ

(119873119898+1

minus 119873119898

)2

119867119898+1

119889x

= intΩ

(ℎ (119873119898

119873119898minus1

) minus 119889119873119873119898

(119867119898+1

minus 119867119898

))

times (119873119898+1

minus 119873119898

) 119889x

(78)

Thus

119889

119889119905

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 2119908119873intΩ

10038161003816100381610038161003816100381610038161003816

(119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

))

times (119873119898+1

minus 119873119898

)

10038161003816100381610038161003816100381610038161003816

119889x

+ 2119889119873intΩ

10038161003816100381610038161003816119873119898

(119867119898+1

minus 119867119898

) (119873119898+1

minus 119873119898

)10038161003816100381610038161003816119889x

le [2119908119873

10038171003817100381710038171003817100381710038171003817

119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

)

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

+2119889119873

10038171003817100381710038171003817119873119898

(119867119898+1

minus 119867119898

)100381710038171003817100381710038171198712(Ω)

]

times10038171003817100381710038171003817119873119898+1

minus 119873119898100381710038171003817100381710038171198712(Ω)

(79)

Next we estimate the above terms

Table 1 Parameter values used in the model

Parameters Range119870119873

5 times 107

cm3

119870119870

5 times 107

cm3

119908119873

1 times 10minus6

s119908119870

1 times 10minus6

s119863119870

2 times 10minus10

cm2s119863119867

5 times 10minus6

cm2s119908119867

22 times 10minus17M sdot cm3s

119889119867

11 times 10minus4

s119889119873

0 rarr 10M sdot s

Let (recall (19))

119872max = max 119872119873119898 =

10038171003817100381710038171198731198981003817100381710038171003817119871infin((0119879]timesΩ)

119873119873119898minus1 =

10038171003817100381710038171003817119873119898minus110038171003817100381710038171003817119871infin((0119879]timesΩ)

(80)

With the embedding constant 1198884= 1198884(Ω 119879)we obtain for the

first term on the right-hand side of (79)

2119908119873

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119873119898

minus 119873119898minus1

minus(119873119898)2

119870119873

+

(119873119898minus1

)2

119870119873

minus119873119898119870119898

119870119870

+119873119898minus1

119870119898minus1

119870119870

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

le 2119908119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+4119908119873119872max119870119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 2119908119873119870119873

119870119870

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

+ 2119908119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

le 119862

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 119862

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

(81)

with 119862= 4119908

119873(1 + (119872max119870119873)) and 119862 = 2119908119873119870119873119870119870

Now for the second term on the right-hand side of (79)

2119889119873

10038171003817100381710038171003817119873119898

(119867119898+1

minus 119867119898

)100381710038171003817100381710038171198712(Ω)

le 1198891198731198884

119870119873

2

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

= 119862

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

(82)


119905 = 1

01

01

1

09

08

07

06

05

04

03

02

01090807060503 04020

119909

(a)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 10

119909

(b)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)

Figure 1 Variations of H+ protons (starred) normal tissue (dashed) and neoplastic tissue (solid) for an aggressive tumor

with 119862= 1198891198731198701198731198884 The two estimates above thus lead to

119889

119889119905

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le1

2(119862

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 119862

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

+119862

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

)

2

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

(83)

Applying Gronwallrsquos inequality we deduce

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198901199052

int

119905

0

(1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

) 119889119904

(84)


01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 1

119909

(a)

01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 10

119909

(b)

01

09

08

07

06

05

04

03

02

001 1

1

090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)

Figure 2 Variations of H+ protons (starred) normal tissue (dashed) and neoplastic tissue (solid) for a less-aggressive tumor comparing tothe one in Figure 1

and finally with119863(Ω 119879) = 1198901198792max1198622 1198622

1198622

we get

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le 119863 (Ω 119879) (10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

)1198795

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+5

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

)

(85)

1198795is chosen such that

119863 (Ω 119879) 1198795le1

4 (86)

Now since 1198701198980 119870119898+1

0isin 1198671(Ω) and 119870

119898(1 minus (119870

119898119870119870))

119870119898+1

(1 minus (119870119898+1

119870119870)) isin 119871

2(0 119879 119871

2(Ω)) we get

119870119898+1

0minus 119870119898

0isin 1198671

(Ω)

[119908119870119870119898+1

(1 minus119870119898+1

119870119870

) minus 119908119870119870119898

(1 minus119870119898

119870119870

)]

isin 1198712

(0 119879 1198712

(Ω))

(87)


055

05

045

04

035

03

025

02

015

01

0050 5 10 15 20 25 30 35 40

119905

119873

120575119873 = 50

120575119873 = 10

120575119873 = 2

(a)

119905 = 10

119873

07

06

05

04

03

02

01

00 01 02 03 04 05 06 07 09 108

120575119873 = 50

120575119873 = 10

120575119873 = 5

120575119873 = 2

120575119873 = 05

(b)

Figure 3 (a) Evolution of the normal cell density for several different values of 120575119873 (b) Normal cell density with respect to the H+ proton

concentration for several different values of 120575119873

119905 = 1

1090807060504030201

00

051 0 02 04

08061

119910

119870

119909

(a)

119905 = 10

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

(b)

119905 = 50

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(c)

119905 = 1

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(d)

119905 = 10

109

08

0706

06

0504

04

0302

02

0100

1 0 02 040806

1

119910

119870

119909

(e)

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

119905 = 50

(f)

Figure 4 Variations of cancer cells for the cases of an aggressive ((a)ndash(c)) and a less-aggressive ((d)ndash(f)) tumor


109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(a)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(b)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(c)

109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(d)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(e)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(f)

Figure 5 Variations of proton concentration for the cases of an aggressive ((a)ndash(c)) and a less-aggressive ((d)ndash(f)) tumor

Theorem 715 in [22] can be applied to the difference119870119898+1minus119870119898 leading to

10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

minus119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

(88)

The right hand side of this inequality can further bemajorizedand with the embedding constants 119888

5= 1198885(Ω 119879) and 119888

6=

1198886(Ω 119879) it follows that

int

119879

0

100381710038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

) minus 119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1199082

119870

1198702

119870

100381710038171003817100381710038171003817(119870119898

)2

minus (119870119898minus1

)2100381710038171003817100381710038171003817

2

1198712(Ω)

+ 1199082

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1198884

5

1199082

119870

1198702

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

10038171003817100381710038171003817119870119898

+ 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

+ 1199082

1198701198882

6

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le (int

119879

0

(41198884

5

1199082

119870

1198702

119870

[10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)

+10038171003817100381710038171003817119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

]

+21199082

1198701198882

6)119889119905)

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le (321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

1198796

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω)

+ 21199082

1198701198882

61198796)

times10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(89)

where

1198796= min 1

81

81205811

8120582

119896 = 321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω) 120582 = 2119908

2

1198701198882

6

(90)


109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(a)

060 02 04

08 106

002

04

04

081

119909

06

002

081

119910

119905 = 10

119873

(b)

109080706

06

0504030201

00

051 0 02 04 08 1

119909119910

119905 = 50

119873

(c)

109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(d)

109080706

06

0504030201

00

005 02 04 08 1119909119910

119905 = 10

119873

(e)

109080706

06

0504030201

100

051 0 02 04 08 1119909

119910

119905 = 50

119873

(f)

Figure 6 Variations of healthy tissue for the cases of an aggressive ((a)ndash(c)) and a less-aggressive ((d)ndash(f)) tumor

Thus putting all together

10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le1

4(310038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

(91)

Therefore (119867119898 119873119898 119870119898) is a Cauchy sequence in 119883 times

119871infin(0 119879 119871

2(Ω)) times 119883 from which the existence of a weak

solution follows

Uniqueness Let (1198701 1198731 1198671) and (119870

2 1198732 1198672) be two solu-

tions to (1)ndash(3) Due to the previous estimates

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883 (92)

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883(93)

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le1

4

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+1

4

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883

+1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883

(94)

thus 1198701= 1198702(92) and with (93) it follows that 119867

1= 1198672

Finally (94) implies that 1198731= 1198732 This completes the proof

of the uniqueness

Regularity of the Solution From (20) it follows that (119870119898 119867119898)is uniformly bounded with respect to119898 in 119884times119884 therefore 119884is compactly embedded in 1198712(0 1198791198671(Ω)) This implies thatfor119898 rarr infin we have (119870119867) isin 119884 times 119884

Theorem 5 (The local solution from Theorem 2 exists glob-ally)

The proof follows upon sequentially extending the timeinterval on which the solution exists the previously deducedestimates allow for a bootstrap of the local existence proof in asubsequent step on the time interval [119879 2119879] then on [2119879 119879]and so forth Eventually the existence of a unique solution inshown on [0 T] for any bounded T

4 Numerical Simulations

In this section we perform the numerical simulation of thesystem (7) The boundary conditions for 119870 and 119867 are theno-flux boundary conditions given by (2) We assume thatinitially the normal cells are at half of their carrying capacitywhile the tumor cells can be close to theirs and thus prone toinvade the surrounding tissue Since the pH level is lowered


by the cancer cells the concentration of protons is takenproportional to the density of the latter Thus using theseassumptionswe choose for the initial conditions of the system(7)

119870 (0 119909) = exp(minus|119909|2

120598) 120598 gt 0

119873 (0 119909) = 05 lowast (1 minus exp(minus|119909|2

120598)) 120598 gt 0

119867 (0 119909) = 120577119870 (0 119909) 120577 isin [0 1)

(95)

where 119909 isin [0 1] and 119909 isin [0 1] times [0 1] in one- and two-dimensional cases respectively In our computations we havetaken both 120577 and the strength parameter 120579 to be 05

For the discretization of the model the finite differencemethod is employed Thereby in the one-dimensional casethe interval [0 1] is divided into 119898 parts with 119898 + 1 nodeswhereas in two dimensions each of the axes is divided into119898 parts thus obtaining a (119898 + 1) times (119898 + 1) mesh in twodimensions Moreover the nodes are reordered in a row-wisemanner leading to a total of (119898 + 1)

2 nodes Thereby thesubindex 119894 in the discretized equations denotes the spatialnode 119909

119894 where 119894 = 1 2 119898 + 1 and 119894 = 1 2 (119898 + 1)

2

in one and two dimensions respectively We use forwarddifferences for the time derivatives in the systemThe centraldifference is used for the diffusion term in the equationcharacterizing the ion concentration

119867119899+1

119894minus 119867119899

119894

Δ119905= 120575119867119870119899

119894minus 120575119867119867119899+1

119894+ (

119867119899+1

119894minus1minus 2119867119899+1

119894+ 119867119899+1

119894+1

Δ1199092)

(96)

where 119899 denotes the time level and Δ119905 and Δ119909 are the timeand space increments respectivelyThe discretized equations(96) for each 119894 leads to the following system of equations ofthe form

AHH119899+1

= H119899 + 120599119899H (97)

with H119899+1 and 120599119899H standing for the vectors containing thevalues of 119867 and Δ119905120575

119867119870 at the (119899 + 1)st and 119899th time levels

at the discretized space points respectively AH being thetridiagonal and block tridiagonalmatrix for the one- and two-dimensional cases respectively The updated values H119899+1 areused to find the values of the normal cell density at the timelevel 119899 + 1 by solving

119873119899+1

119894=

1

1 + Δ119905120575119873119867119899+1

119894

[119873119899

119894+ Δ119905119873

119899

119894(1 minus 120579119870

119899

119894minus 119873119899

119894)] (98)

for each space point 119894In order to write the term characterizing the dispersion

of the neoplastic tissue into the healthy tissue we make use ofthe nonstandard finite difference scheme [24] that is

nabla (119863 (119873)nabla119870)|119909119894=

1

2(Δ119909)2sum

119896isin119873119894

(119863 (119873119899+1

119896) + 119863 (119873

119899+1

119894))

times (119870119899+1

119896minus 119870119899+1

119894)

(99)

where119863(119873) = 120575119896(1 minus119873) and119873

119894sub 119868 is the index set pointing

at the direct neighbours of 119909119894on the grid Thus 119873

119894has two

elements for the one-dimensional case and four elements forthe two-dimensional case and the matrix-vector form of thescheme reads

AKK119899+1

= K119899 + 120599K (100)

where AK is the (119898 + 1) times (119898 + 1) tridiagonal matrix or theblock tridiagonalmatrix of size (119898+1)2times(119898+1)2 for the one-and two-dimensional cases respectively coming from thenonstandard finite difference discretization 120599K is the vectorcoming from the proliferation term with the entries 120588

119896119870119899

119894(1minus

119870119899

119894minus 119873119899+1

119894) and K119899+1 and K119899 are the vectors containing the

119870 values at the discretized points for the time levels 119899+ 1 and119899 respectively

Throughout our simulations we use the biological param-eter values from Table 1 which is reproduced from [24]

According to a linear stability analysis performed in [24]the parameter 120575

119873plays a crucial role in characterizing the

aggressivity of the tumor There 120575119873= 1 was shown to be

the crossover value for 120575119873lt 1 the tumor is less aggressive

whereas for 120575119873gt 1 it becomes highly aggressive This will be

an important factor in our computations too

41 The One-Dimensional Case We consider the space inter-val [0 1] and choose for the time and space increments Δ119905 =01 and Δ119909 = 001 respectively

In Figures 1 and 2 we present the simulations with 120575119873=

50 (an aggressive tumor) and 120575119873= 05 (a less-aggressive

one) respectively The rest of parameters are taken fromTable 1 As time progresses the difference between these twocases is more visible In the aggressive case at later times thecancer cells invade a larger region and destroy the healthytissue much more than in the case of a less-aggressive tumor(Figure 2)

In the next set of graphs we consider the negative effect ofthe aggressivity parameter 120575

119873on the normal cell density We

know from the nondimensionalization addressed in Section 2that the (nondimensionalized) parameter 120575

119873is proportional

to the death rate 119889119873and inversely proportional to the proton

reabsorption rate 119889119867

Figure 3(a) shows the evolution of the normal cell densitywith respect to 120575

119873for the times up to 119905 = 40 at a fixed

space point 119909 = 04 One can notice that a more aggressivetumor (larger 120575

119873or equivalently larger 119889

119873) leads to a faster

decay in the density of the normal cells as expected On theother hand Figure 3(a) also supports the intuitive fact thatin an organism which can poorly buffer the issuing excessiveprotons (smaller 119889

119867and larger 120575

119873 resp) the pH value will

decay faster as well thus triggering the decay of normal celldensity which in such an acid environment can no longer besustained at a physiologically convenient level

Figure 3(b) illustrates the normal cell density at 119905 = 10

depending on the concentration of H+ protons for severaldifferent values of 120575

119873 In an organism whose normal cells

are more sensitive to pH variations (larger 119889119873) the density

of these cells will decay faster for the same concentration ofH+ protons It can also be seen that a smaller reabsorption


rate of excessive protons leads to a faster decay of normal celldensity

42The Two-Dimensional Case We perform 2D simulationsin the unit square [0 1] times [0 1] using 120575

119867= 70 and still

with the parameter values in Table 1 Analogously to ourcomputations in 1D we consider two different cases 120575

119873=

125 (an aggressive tumor) and 120575119873

= 05 lt 1 (a less-aggressive one) We use Δ119905 = 001 as the time increment andfor the spatial discretization we take 11 nodes on each axisleading to 121 nodes in the computational domain

The evolution of cancer cells is plotted in Figure 4 Atan earlier time (eg 119905 = 1 see Figures 4(a) and 4(d)) thedifference between the less- and the higher-aggressive tumorsis not relevant However as time progresses the differencestarts to be visible (eg at 119905 = 10 see Figures 4(b) and 4(e))whereas by 119905 = 50 the more-aggressive tumor (Figure 4(c))invaded almost the whole domain and the less-aggressive onewas not able to penetrate that far

Also observe that the proton concentration varies propor-tionally to the tumor cell density and is inversely proportionalto the normal cell density (Figures 5 and 6 resp) Moreoverthe healthy tissue is completely destroyed in the case of anaggressive tumor (Figure 6(c)) by 119905 = 50

Acknowledgments

C Surulescu acknowledges the support of the Baden-Wurttemberg Foundation G Meral acknowledges the sup-port of LLP Erasmus Staff Mobility Programme during hervisit to the University of Kaiserslautern

References

[1] A Ashkenazi and V M Dixit ldquoDeath receptors signaling andmodulationrdquo Science vol 281 no 5381 pp 1305ndash1308 1998

[2] H Yamaguchi F Pixley and J Condeelis ldquoInvadopodia andpodosomes in tumor invasionrdquoEuropean Journal of Cell Biologyvol 85 no 3-4 pp 213ndash218 2006

[3] J Kelkel and C Surulescu ldquoOn some models for cancer cellmigration through tissue networksrdquo Mathematical Biosciencesand Engineering vol 8 no 2 pp 575ndash589 2011

[4] J Kelkel and C Surulescu ldquoAmultiscale approach to cell migra-tion in tissue networksrdquo Mathematical Models and Methods inApplied Sciences vol 22 no 3 Article ID 1150017 25 pages 2012

[5] T Hillen ldquoM5 mesoscopic and macroscopic models for mes-enchymal motionrdquo Journal of Mathematical Biology vol 53 no4 pp 585ndash616 2006

[6] A Chauviere T Hillen and L Preziosi ldquoModeling cell move-ment in anisotropic and heterogeneous network tissuesrdquo Net-works and Heterogeneous Media vol 2 no 2 pp 333ndash357 2007

[7] A R A Anderson M A J Chaplain E L Newman R J CSteele and AMThompson ldquoMathematical modeling of tumorinvasion and metastasisrdquo Journal of Theoretical Medicine vol 2pp 129ndash154 2000

[8] R A Gatenby and E T Gawlinski ldquoA reaction-diffusion modelof cancer invasionrdquo Cancer Research vol 56 no 24 pp 5745ndash5753 1996

[9] R A Gatenby and R J Gillies ldquoGlycolysis in cancer a potentialtarget for therapyrdquo International Journal of Biochemistry andCellBiology vol 39 no 7-8 pp 1358ndash1366 2007

[10] D Hanahan and R AWeinberg ldquoHallmarks of cancer the nextgenerationrdquo Cell vol 144 no 5 pp 646ndash674 2011

[11] H Izumi T Torigoe H Ishiguchi et al ldquoCellular pH regulatorspotentially promising molecular targets for cancer chemother-apyrdquoCancer Treatment Reviews vol 29 no 6 pp 541ndash549 2003

[12] O R Abakarova ldquoThe metastatic potential of tumors dependson the pH of host tissuesrdquo Bulletin of Experimental Biology andMedicine vol 120 no 6 pp 1227ndash1229 1995

[13] R Martınez-Zaguilan E A Seftor R E B Seftor Y W Chu RJ Gillies andM J CHendrix ldquoAcidic pH enhances the invasivebehavior of human melanoma cellsrdquo Clinical and ExperimentalMetastasis vol 14 no 2 pp 176ndash186 1996

[14] R A Gatenby and E T Gawlinski ldquoThe glycolytic phenotypein carcinogenesis and tumor invasion insights through mathe-matical modelsrdquoCancer Research vol 63 no 14 pp 3847ndash38542003

[15] A Fasano M A Herrero and M R Rodrigo ldquoSlow and fastinvasion waves in a model of acid-mediated tumour growthrdquoMathematical Biosciences vol 220 no 1 pp 45ndash56 2009

[16] K Smallbone D J Gavaghan R A Gatenby and P K MainildquoThe role of acidity in solid tumour growth and invasionrdquoJournal ofTheoretical Biology vol 235 no 4 pp 476ndash484 2005

[17] L Bianchini and A Fasano ldquoAmodel combining acid-mediatedtumour invasion and nutrient dynamicsrdquo Nonlinear AnalysisReal World Applications vol 10 no 4 pp 1955ndash1975 2009

[18] J Kelkel and C Surulescu ldquoA weak solution approach toa reaction-diffusion system modeling pattern formation onseashellsrdquoMathematicalMethods in the Applied Sciences vol 32no 17 pp 2267ndash2286 2009

[19] J Kelkel and C Surulescu ldquoOn a stochastic reaction-diffusionsystem modeling pattern formation on seashellsrdquo Journal ofMathematical Biology vol 60 no 6 pp 765ndash796 2010

[20] A S Silva J A Yunes R J Gillies and R A Gatenby ldquoThepotential role of systemic buffers in reducing intratumoralextracellular pH and acid-mediated invasionrdquo Cancer Researchvol 69 no 6 pp 2677ndash2684 2009

[21] I F Tannock and D Rotin ldquoAcid pH in tumors and its potentialfor therpeutic exploitationrdquo Cancer Research vol 49 no 16 pp4373ndash4384 1989

[22] L C Evans Partial Differential Equations vol 19 AmericanMathematical Society Providence RI USA 1998

[23] A D Polyanin Handbook of Linear Partial Differential Equa-tions for Engineers and Scientists Chapman amp Hall New YorkNY USA 2002

[24] H J Eberl and L Demaret ldquoA finite difference scheme for adegenerated diffusion equation arising in microbial ecologyrdquoElectronic Journal of Differential Equations vol 15 pp 77ndash952007

Submit your manuscripts athttpwwwhindawicom

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Algebra

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Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Hence the pH level directly influences the metastatic poten-tial of tumor cells [12 13] These facts led Gatenby andGawlinski [8 14] to propose a model for the acid-mediatedtumor invasion which describes the interaction between thedensity of normal cells tumor cells and the concentrationof H+ protons produced by the latter via reaction-diffusionequations Starting from this model travelling waves havebeen used to explain the aggressive action of cancer cells ontheir surroundings [15] Further settings issued fromGatenbyand Gawlinskirsquos model involve nutrient dynamics influencedby both vascular and avascular growth of multicellular tumorspheroids [16 17] assuming rotational symmetry and investi-gating existence and qualitative properties of the solutions

In this work we reconsider the model in [8] whereby alsoexplicitly allowing for crowding effects (due to competitionwith cancer cells) in the growth of normal cells (Thiscan be also done for the growth term in the tumor cellequation however it does not change the analysis nor thequalitative behavior of the solutions to the system Moreoverits biological motivation is not strong enough since thecompetition between the two cell types does not reallyaffect the growth but rather the invasion of the neoplastictissue the acidity increase in the peritumoral environmentis a byproduct of the enhanced glycolysis of cancer cellsand not produced with the purpose of killing the normalcells and eluding concurrence) For this setting we performmathematical analysis and numerical simulations in order toverify the model predictions with respect to experimentallyobserved qualitative facts In order to prove the existence ofa unique (weak) solution we propose an intuitive methodrelying on an iterative procedure which has also been appliedin [18 19] in a different context (one of the supplementarydifficulties here is the diffusion coefficient being nonconstantbut depending on the solution itself) and allows to avoid theuse of operator semigroups

2 Problem Setting

The model by Gatenby and Gawlinski [8] describes theevolution of normal and tumor cell density respectively ina domain where these cell types interact on the basis of pHvalue modifications The mathematical description of theseprocesses is ensured by the following system of reaction-diffusion equations for the normal cell density 119873(119905 x) thetumor cell density 119870(119905 x) (both in cellscm3) and theconcentration of excessive H+(119905 x) ion concentration (inMol)

120597119873

120597119905= 119908119873119873(1 minus

119873

119870119873

minus 120579119870

119870119870

) minus 119889119873119867119873 in (0 119879) times Ω

120597119870

120597119905= 119908119870119870(1 minus

119870

119870119870

) + nabla sdot (119863119870(1 minus

119873

119870119873

)nabla119870)

in (0 119879) times Ω

120597119867

120597119905= 119908119867119870 minus 119889

119867119867 + 119863

119867Δ119867 in (0 119879) times Ω

(1)

where 120579 le 12 denotes the strength parameter for thecompetition between normal cells and neoplastic tissueThereby Ω sub R119899 (119899 = 1 2 3) is a regular-enough andbounded domain and only microscopically small processesare considered at the interface between tumor and healthytissue Observe that the diffusion coefficient of the cancercells depends on the normal cell density when the healthytissue is at its carrying capacity the neoplastic tissue cannotdiffuse thus the tumor is confined It can only spread if thesurrounding normal tissue is diminished from its carryingcapacity and this is assumed to happen due to lowering thepH level upon secretion of H+ protons by cancer cells

The constants 119908119873and 119908

119870are given in 1s and represent

the growth rates and the constants119870119873and119870

119870are expressed

in cellscm3 and provide the carrying capacities of the normaland tumor cells respectivelyThe death rate 119889

119873of the normal

cells is measured in 1(Mol sdot s) and the diffusion coefficientof tumor cells in the absence of normal cells is given incm2s The production rate 119908

119867of H+ protons is expressed

inMol sdot cm3(cells sdot s) the uptake rate 119889119867(due for instance to

proton buffering (see eg [20] and the references therein)andor various ion exchangers between intracellular andextracellular domains (see eg [21])) is measured in 1s andtheir diffusion coefficient of119863

119867in cm2s

We also assume that there is no exchange of cells andH+ protons through the boundary of the considered domainthus

120597119870

120597n=120597119867

120597n= 0 in (0 119879) times 120597Ω (2)

where n denotes the outer unit normal vector to 120597ΩThe initial conditions are given by

119870 (0 x) = 1198700(x) 119873 (0 x) = 119873

0(x)

119867 (0 x) = 1198670(x) in Ω

(3)

Thereby 1198700(x) 119873

0(x) and 119867

0(x) are strictly positive func-

tions which satisfy the no-flux condition

1205971198700

120597n=1205971198670

120597n= 0 on 120597Ω (4)

In order to render the system (1) dimensionless we use thefollowing transformations

=119873

119870119873

=119870

119870119870

= 119867 sdot119889119867

119908119867119870119870

= 119908119873sdot 119905 x = radic

119908119873

119863119867

sdot x(5)

along with the notations

120575119873=119889119873119908119867119870119870

119889119867119908119873

120588119870=119908119870

119908119873

Δ119870=119863119870

119863119867

120575119867=119889119867

119908119873

(6)



120597119873


119873119867119873

120597119870


119870(1 minus 119873)nabla119870)

120597119867

120597119905= 120575119867119870 minus 120575

119867119867 + Δ119867

(7)





119883 = 119871infin

(0 1198791198671

(Ω))

119884 = 119906 isin 1198712

(0 1198791198672

(Ω)) 119906119905isin 1198712

(0 119879 1198712

(Ω))

119885 = 119871infin

(0 119879 1198712

(Ω))

(8)


intΩ

119908119867119870120601119889x

= intΩ

119867119905120601119889x + int

Ω

119863119867nabla119867nabla120601119889x + int

Ω

119889119867119867120601119889x

intΩ

119908119873119873(1 minus

119873

119870119873

minus 120579119870

119870119870

)120601119889x

= intΩ

119873119905120601119889x + int

Ω

119873119889119873119867120601119889x

intΩ

119908119870119870(1 minus

119870

119870119870

)120601119889x

= intΩ

119870119905120601119889x + int

Ω

119863119870(1 minus

119873

119870119873

)nabla119870nabla120601119889x

(9)


1198670isin 1198671

(Ω) cap 119862 (Ω) 1198730isin 119871infin

(Ω) cap 1198671

(Ω)

1198700isin 1198671

(Ω)

1198670ge 119862119867gt 0 0 lt 119873

0le119870119873

2 0 lt 119870

0le 119870119870

(10)


We set

119879 =

6

prod

119894=1

119879119894

(11)



(119867119898

119870119898

)119898isinN0


(119873119898

)119898isinN0

isin 119885

(12)



1205971198670

120597119905minus 119863119867Δ1198670

+ 1198891198671198670

= 0 (13)

1205971198730

120597119905+ 1198891198731198670

1198730

= 0 (14)

1205971198700

120597119905minus nabla sdot (119863

119870(1 minus

1198730

119870119873

)nabla1198700

) = 0 (15)

while (119867119898 119870119898)119898isinN0


isin 119885


120597119867119898+1

120597119905minus 119863119867Δ119867119898+1

+ 119889119867119867119898+1

= 119908119867119870119898

(16)

120597119873119898+1

120597119905+ 119889119873119867119898+1

119873119898+1

= 119908119873119873119898

(1 minus119873119898

119870119873

minus 120579119870119898

119870119870

)

(17)

120597119870119898+1

120597119905minus nabla sdot (119863

119870(1 minus

119873119898+1

119870119873

)nabla119870119898+1

)

= 119908119870119870119898

(1 minus119870119898

119870119870

)

(18)






0it holds that

119873119898

119873119898


((0 119879] times Ω) (19)

119867119898

119870119898

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

119867119898

119905 119870119898

119905isin 1198712

(0 119879 1198712

(Ω))

(20)




119867119898

(119905 x) ge 119862119867119890minus119889119867119905 119873

119898

(119905 x) le 119870119873

2

119870119898

(119905 x) le 119870119870



0the estimates

10038171003817100381710038171198671198981003817100381710038171003817119883

+100381710038171003817100381711986711989810038171003817100381710038171198712(01198791198672(Ω))

le 119862 (Ω 119879) (10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)+10038171003817100381710038171198670

10038171003817100381710038171198671(Ω))

(22)

10038171003817100381710038171198731198981003817100381710038171003817

2

119883le 119862 (Ω 119879)

100381710038171003817100381711987301003817100381710038171003817

2

1198671(Ω) (23)

10038171003817100381710038171198701198981003817100381710038171003817119883

+100381710038171003817100381711987011989810038171003817100381710038171198712(01198791198672(Ω))

le 2119862 (Ω 119879)10038171003817100381710038171198700

10038171003817100381710038171198671(Ω) (24)


119873119898

isin 119871infin

(0 119879 1198712

(Ω)) (25)





0

(119905 x) = 1198670 (119905 x) 119890119889119867119905 (26)


0

119905minus 119863119867Δ0

= 0 (27)


0isin 1198671(Ω) it


1198670

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

1198670

119905isin 1198712

(0 119879 1198712

(Ω))

(28)


+100381710038171003817100381710038171198670100381710038171003817100381710038171198712(01198791198672(Ω))

le 119862 (Ω 119879)10038171003817100381710038171198670

10038171003817100381710038171198671(Ω) (29)


1198730

(119905 x) = 1198730119890minusint

119905

01198891198731198670(119904x)119889119904

gt 0 (30)


10038171003817100381710038171003817119873010038171003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

=

10038171003817100381710038171003817100381710038171198730119890minusint

119905

01198891198731198670119889119905

1003817100381710038171003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

le10038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

(31)



show first that

1198730

isin 119871infin

((0 119879] times Ω) (32)

1198730


((0 119879] times Ω) (33)


(30)

=

10038171003817100381710038171003817100381710038171198730119890minusint

119905

01198891198731198670119889119905

1003817100381710038171003817100381710038171003817119871infin((0119879]timesΩ)

le10038171003817100381710038171198730

1003817100381710038171003817119871infin(Ω)lt infin

(34)

For 119905 ge 120575 gt 0 it is100381710038171003817100381710038171198730

119905

10038171003817100381710038171003817119871infin((0119879]timesΩ)

(30)

= 119889119873

10038171003817100381710038171003817100381710038171198730sdot 119890minusint

119905

01198891198731198670119889119905

sdot 1198670


(35)



0


Therefore


1198670

(119905 x) = lim(119905x)rarr (0x0)

0

(119905 x) 119890minus119889119867119905 = 1198670(x0)

(37)



119870(1 minus (119873

0(119905 x)119870

119873)) in (15)

depends on timeLet

119896119898(119905) =

119898

sum

119894=1

119889119894

119898(119905) 119908119894

(38)


119908119894infin


119908119894infin


(39)


119860 [119896119898 119896119898] = intΩ

119886 (119905 x) (nabla119896119898)2

119889x (40)



119889

119889119905119860 [119896119898 119896119898] = intΩ

1198861015840

(119905 x) (nabla119896119898)2

119889x

+ 2intΩ

119886 (119905 x) (nabla119896119898)1015840

nabla119896119898119889x

(41)


intΩ

1198861015840

(119905 x) (nabla119896119898)2


119863119870

119870119873

(1198730

)1015840

(119905 x) (nabla119896119898)2

119889x

(42)



100381710038171003817100381710038171198961015840

119898

10038171003817100381710038171003817

2

1198712(Ω)

+119889

119889119905(1

2119860 [119896119898 119896119898])

le119862

120577(1003817100381710038171003817119896119898

1003817100381710038171003817

2

1198671(Ω)

+10038171003817100381710038171198911003817100381710038171003817

2

1198712(Ω))

+ 2120577100381710038171003817100381710038171198961015840

119898

10038171003817100381710038171003817

2

1198712(Ω)

+1

2intΩ

119863119870

119870119873

(1198730

)1015840

(nabla119896119898)2

119889x

(43)


1198721198730 =

119863119870

119870119873

100381710038171003817100381710038171198730

119905

10038171003817100381710038171003817119871infin((0119879]timesΩ) (44)


int

119879

0

intΩ

119863119870

119870119873

(1198730

)1015840

(nabla119896119898)2

119889x 119889119905 le 1198721198730 int

119879

0

1003817100381710038171003817nabla1198961198981003817100381710038171003817

2

1198712(Ω)119889119905

le 1198721198730

10038171003817100381710038171198961198981003817100381710038171003817

2

1198712(0119879119867

1(Ω))

le 120574 (Ω 119879) lt infin

(45)




1198700

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

1198700

119905isin 1198712

(0 119879 1198712

(Ω))

10038171003817100381710038171003817119870010038171003817100381710038171003817119883

+100381710038171003817100381710038171198700100381710038171003817100381710038171198712(01198791198672(Ω))

le 119862 (Ω 119879)10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)

(46)

Since

1198700

0(x) gt 0 (47)





0


1= 1198881(Ω 119879)

int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198712(Ω)119889119905 le 119888

1int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)119889119905

ind hyple(24)

411988811198622

(Ω 119879) 11987910038171003817100381710038171198700

1003817100381710038171003817

2

1198671(Ω)

lt infin

(48)

and thus

119870119898

isin 1198712

(0 119879 1198712

(Ω)) (49)


119867119898+1

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

119867119898+1

119905isin 1198712

(0 119879 1198712

(Ω))

10038171003817100381710038171003817119867119898+110038171003817100381710038171003817119883

+10038171003817100381710038171003817119867119898+1100381710038171003817100381710038171198712(01198791198672(Ω))

le 1198621(Ω 119879) (2119908

119867119862 (Ω 119879)radic119888

111987910038171003817100381710038171198700

10038171003817100381710038171198671(Ω)+10038171003817100381710038171198670

10038171003817100381710038171198671(Ω))

le C (Ω 119879) (10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)+10038171003817100381710038171198670

10038171003817100381710038171198671(Ω))

(50)

withC(Ω 119879) = max1198621(Ω 119879) 119862

1(Ω 119879)2119908

119867119862(Ω 119879)radic119888

1119879


119898+1(119905 x) minus 119862

119867119890minus119889119867119905 for

which it holds

⟨120595119898+1

119905(119905) 120601⟩ + 119863

119867intΩ

nabla120595119898+1

nabla120601119889x + 119889119867intΩ

120595119898+1

120601119889x

= ⟨119908119867119870119898

120601⟩ (51)



119867119890minus119889119867119905


119873119898+1

(119905 x) = 119890minus120572(119905x) (1198730(x) + int

119905

0

120573 (119904 x) 119890120572(119904x)119889119904) (52)


where

120572 (119905 x) = int119905

0

119889119873119867119898+1

(V x) 119889V

120573 (119904 x) = 119908119873119873119898

(119904 x) (1 minus 119873119898

(119904 x)119870119873

minus 120579119870119898

(119904 x)119870119870

)

(53)


119873119898+1

isin 119871infin

((0 119879] times Ω) (54)

119873119898+1


((0 119879] times Ω) (55)


Next estimate10038171003817100381710038171003817119873119898+1

119905

10038171003817100381710038171003817119871infin((0119879]timesΩ)

le 119908119873

10038171003817100381710038171198731198981003817100381710038171003817119871infin((0119879]timesΩ)

10038171003817100381710038171003817100381710038171003817

1 minus119873119898

119870119873

minus 120579119870119898

119870119870

10038171003817100381710038171003817100381710038171003817119871infin((0119879]timesΩ)

+ 119889119873

10038171003817100381710038171003817119873119898+110038171003817100381710038171003817119871infin((0119879]timesΩ)

10038171003817100381710038171003817119867119898+110038171003817100381710038171003817119871infin((0119879]timesΩ)

lt infin

(56)



119871infin((0119879]timesΩ)


10038171003817100381710038171003817119873119898+1

119905




119873119898+1

(119905 x) le 119870119873

2119890minus120572(119905x)

+ int

119905

0

120573 (119904 x) 119890minus(120572(119905x)minus120572(119904x))119889119904

le119870119873

2119890minus120572(119905x)

+ 119908119873

119870119873

2int

119905

0

119890minus(120572(119905x)minus120572(119904x))

119889119904

(58)


119867119898+1


estimate

119873119898+1

(119905 x) le 119870119873

2119890minus119889119873119867119905 + 119908

119873

119870119873

2

1

119889119873119867

(1 minus 119890minus119889119873119867119905)

le119870119873

2((1 minus

119908119873

119889119873119867

) 119890minus119889119873119867119905 +

119908119873

119889119873119867

) le119870119873

2

(59)


Due to (52) we get

10038171003817100381710038171003817119873119898+1

(119905)10038171003817100381710038171003817

2

1198671(Ω)

=

10038171003817100381710038171003817100381710038171003817

119890minus120572(119905)

1198730+ 119890minus120572(119905)

int

119905

0

120573(119904)119890120572(119904)

119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 21199082

119873

10038171003817100381710038171003817100381710038171003817

int

119905

0

119873119898

(119904) (1 minus119873119898(119904)

119870119873

minus 120579119870119898(119904)

119870119870

)119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 21199082

119873

100381710038171003817100381710038171003817100381710038171003817

int

119905

0

(119873119898

(119904) minus(119873119898(119904))2

1198702

119873

)119889119904

100381710038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 41199082

119873(

10038171003817100381710038171003817100381710038171003817

int

119905

0

119873119898

(119904)119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

+1

1198702

119873

10038171003817100381710038171003817100381710038171003817

int

119905

0

(119873119898

(119904))2

119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

)

le10038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)[2 + 4119908

2

119873119862 (Ω 119879) 119879

2

]

le C (Ω 119879)10038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

(60)




int

119879

0

10038171003817100381710038171003817100381710038171003817

119870119898

(1 minus119870119898

119870119870

)

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le int

119879

0

(100381710038171003817100381711987011989810038171003817100381710038171198712(Ω)

+

1003817100381710038171003817100381710038171003817100381710038171003817

(119870119898)2

119870119870

10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

)

2

119889119905

le 2int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198712(Ω)119889119905 + 2int

119879

0

1003817100381710038171003817100381710038171003817100381710038171003817

(119870119898)2

119870119870

1003817100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 21198882

1int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)119889119905 + 2

1198884

2

1198702

119870

int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

4

1198671(Ω)119889119905

le 81198882

11198622

(Ω 119879)10038171003817100381710038171198700

1003817100381710038171003817

2

1198671(Ω)11987911198792

+ 321198884

2

1198702

119870

1198624

(Ω 119879)10038171003817100381710038171198700

1003817100381710038171003817

4

1198671(Ω)11987911198792lt infin

(61)


2(0 119879 119871

2(Ω)) and119870




119870119898+1

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

119870119898+1

119905isin 1198712

(0 119879 1198712

(Ω))

(62)


11198622(Ω 119879) 119879

11198624(Ω 119879) le 1

and

1198792= min1

2

1

161199082

1198701198882

1

100381710038171003817100381711987001003817100381710038171003817

1198702

119870

641199082

1198701198884

2

100381710038171003817100381711987001003817100381710038171003817

3 (63)

Then

int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905 le10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)(64)


+10038171003817100381710038171003817119870119898+1100381710038171003817100381710038171198712(01198791198672(Ω))

le 2119862 (Ω 119879)10038171003817100381710038171198700

10038171003817100381710038171198671(Ω) (65)


auxiliary function

120585119898+1


(119905 x) (66)


0

119870119898+1

le 119870119870 (67)





0


120597120585119898+1

120597119905minus nabla sdot (119863

119870(1 minus

119873119898+1

119870119873

)nabla120585119898+1

)


(1 minus119870119898

119870119870

)

(68)

Since

119870119898

(1 minus119870119898

119870119870

) le 119870119870 (69)



(1 minus119870119898

119870119870

) ge 0 (70)


3and 120578 such




(119905 x) = 120585119898+1 ge 0 (71)


119870119898+1

le 119870119870 (72)





0and





2(Ω) this



0 119867119898+1

0isin 1198671(Ω)


119867119898+1

0minus 119867119898

0isin 1198671

(Ω)

119870119898+1

minus 119870119898

isin 1198712

(0 119879 1198712

(Ω))

(73)



10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817119908119867119870119898

minus 119908119867119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

(74)


3= 1198883(Ω 119879) it follows that

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879)1199082

1198671198882

3int

119879

0

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le 119862 (Ω 119879)1199082

1198671198882

31198794

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

2

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(75)

where

1198794= min1

4

1

4119862 (Ω 119879)1199082

1198671198882

3

(76)




120597

120597119905(119873119898+1

minus 119873119898

) + 119889119873(119867119898+1

119873119898+1

minus 119867119898

119873119898

)

= 119908119873(119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

))

(77)


119898119870119873)minus(119870

119898119870119870))minus

119873119898minus1

(1 minus (119873119898minus1

119870119873) minus (119870

119898minus1119870119870)))




1

2intΩ

120597

120597119905(119873119898+1

minus 119873119898

)2

119889x

+ 119889119873intΩ

(119873119898+1

minus 119873119898

)2

119867119898+1

119889x

= intΩ

(ℎ (119873119898

119873119898minus1

) minus 119889119873119873119898

(119867119898+1

minus 119867119898

))

times (119873119898+1

minus 119873119898

) 119889x

(78)

Thus

119889

119889119905

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 2119908119873intΩ

10038161003816100381610038161003816100381610038161003816

(119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

))

times (119873119898+1

minus 119873119898

)

10038161003816100381610038161003816100381610038161003816

119889x

+ 2119889119873intΩ

10038161003816100381610038161003816119873119898

(119867119898+1

minus 119867119898

) (119873119898+1

minus 119873119898

)10038161003816100381610038161003816119889x

le [2119908119873

10038171003817100381710038171003817100381710038171003817

119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

)

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

+2119889119873

10038171003817100381710038171003817119873119898

(119867119898+1

minus 119867119898

)100381710038171003817100381710038171198712(Ω)

]

times10038171003817100381710038171003817119873119898+1

minus 119873119898100381710038171003817100381710038171198712(Ω)

(79)




5 times 107

cm3

119870119870

5 times 107

cm3

119908119873

1 times 10minus6

s119908119870

1 times 10minus6

s119863119870

2 times 10minus10

cm2s119863119867

5 times 10minus6

cm2s119908119867


119889119867

11 times 10minus4

s119889119873

0 rarr 10M sdot s

Let (recall (19))

119872max = max 119872119873119898 =

10038171003817100381710038171198731198981003817100381710038171003817119871infin((0119879]timesΩ)

119873119873119898minus1 =


(80)



2119908119873

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119873119898


minus(119873119898)2

119870119873

+

(119873119898minus1

)2

119870119873

minus119873119898119870119898

119870119870

+119873119898minus1

119870119898minus1

119870119870

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

le 2119908119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+4119908119873119872max119870119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 2119908119873119870119873

119870119870

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

+ 2119908119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

le 119862

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 119862

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

(81)

with 119862= 4119908

119873(1 + (119872max119870119873)) and 119862 = 2119908119873119870119873119870119870


2119889119873

10038171003817100381710038171003817119873119898

(119867119898+1

minus 119867119898

)100381710038171003817100381710038171198712(Ω)

le 1198891198731198884

119870119873

2

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

= 119862

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

(82)


119905 = 1

01

01

1

09

08

07

06

05

04

03

02

01090807060503 04020

119909

(a)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 10

119909

(b)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)



119889

119889119905

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le1

2(119862

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 119862

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

+119862

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

)

2

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

(83)


10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198901199052

int

119905

0

(1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

) 119889119904

(84)


01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 1

119909

(a)

01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 10

119909

(b)

01

09

08

07

06

05

04

03

02

001 1

1

090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)



1198622

we get

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le 119863 (Ω 119879) (10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

)1198795

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+5

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

)

(85)


119863 (Ω 119879) 1198795le1

4 (86)

Now since 1198701198980 119870119898+1

0isin 1198671(Ω) and 119870

119898(1 minus (119870

119898119870119870))

119870119898+1

(1 minus (119870119898+1

119870119870)) isin 119871

2(0 119879 119871

2(Ω)) we get

119870119898+1

0minus 119870119898

0isin 1198671

(Ω)

[119908119870119870119898+1

(1 minus119870119898+1

119870119870

) minus 119908119870119870119898

(1 minus119870119898

119870119870

)]

isin 1198712

(0 119879 1198712

(Ω))

(87)


055

05

045

04

035

03

025

02

015

01

0050 5 10 15 20 25 30 35 40

119905

119873

120575119873 = 50

120575119873 = 10

120575119873 = 2

(a)

119905 = 10

119873

07

06

05

04

03

02

01

00 01 02 03 04 05 06 07 09 108

120575119873 = 50

120575119873 = 10

120575119873 = 5

120575119873 = 2

120575119873 = 05

(b)



119905 = 1

1090807060504030201

00

051 0 02 04

08061

119910

119870

119909

(a)

119905 = 10

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

(b)

119905 = 50

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(c)

119905 = 1

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(d)

119905 = 10

109

08

0706

06

0504

04

0302

02

0100

1 0 02 040806

1

119910

119870

119909

(e)

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

119905 = 50

(f)



109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(a)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(b)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(c)

109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(d)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(e)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

minus119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

(88)


5= 1198885(Ω 119879) and 119888

6=


int

119879

0

100381710038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

) minus 119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1199082

119870

1198702

119870

100381710038171003817100381710038171003817(119870119898

)2


)2100381710038171003817100381710038171003817

2

1198712(Ω)

+ 1199082

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1198884

5

1199082

119870

1198702

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

10038171003817100381710038171003817119870119898

+ 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

+ 1199082

1198701198882

6

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le (int

119879

0

(41198884

5

1199082

119870

1198702

119870

[10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)

+10038171003817100381710038171003817119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

]

+21199082

1198701198882

6)119889119905)

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le (321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

1198796

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω)

+ 21199082

1198701198882

61198796)

times10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(89)

where

1198796= min 1

81

81205811

8120582

119896 = 321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω) 120582 = 2119908

2

1198701198882

6

(90)


109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(a)

060 02 04

08 106

002

04

04

081

119909

06

002

081

119910

119905 = 10

119873

(b)

109080706

06

0504030201

00

051 0 02 04 08 1

119909119910

119905 = 50

119873

(c)

109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(d)

109080706

06

0504030201

00

005 02 04 08 1119909119910

119905 = 10

119873

(e)

109080706

06

0504030201

100

051 0 02 04 08 1119909

119910

119905 = 50

119873

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le1

4(310038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

(91)


119871infin(0 119879 119871


solution follows


2 1198732 1198672) be two solu-


10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883 (92)

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883(93)

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le1

4

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+1

4

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883

+1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883

(94)


1= 1198672


of the uniqueness








119870 (0 119909) = exp(minus|119909|2

120598) 120598 gt 0


120598)) 120598 gt 0

119867 (0 119909) = 120577119870 (0 119909) 120577 isin [0 1)

(95)




119894 where 119894 = 1 2 119898 + 1 and 119894 = 1 2 (119898 + 1)

2


119867119899+1

119894minus 119867119899

119894

Δ119905= 120575119867119870119899

119894minus 120575119867119867119899+1

119894+ (

119867119899+1

119894minus1minus 2119867119899+1

119894+ 119867119899+1

119894+1

Δ1199092)

(96)


AHH119899+1

= H119899 + 120599119899H (97)




119873119899+1

119894=

1

1 + Δ119905120575119873119867119899+1

119894

[119873119899

119894+ Δ119905119873

119899

119894(1 minus 120579119870

119899

119894minus 119873119899

119894)] (98)



nabla (119863 (119873)nabla119870)|119909119894=

1

2(Δ119909)2sum

119896isin119873119894

(119863 (119873119899+1

119896) + 119863 (119873

119899+1

119894))

times (119870119899+1

119896minus 119870119899+1

119894)

(99)

where119863(119873) = 120575119896(1 minus119873) and119873



119894has two


AKK119899+1

= K119899 + 120599K (100)


119896119870119899

119894(1minus

119870119899

119894minus 119873119899+1


























119867and larger 120575













119873=





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120597119873


119873119867119873

120597119870


119870(1 minus 119873)nabla119870)

120597119867

120597119905= 120575119867119870 minus 120575

119867119867 + Δ119867

(7)





119883 = 119871infin

(0 1198791198671

(Ω))

119884 = 119906 isin 1198712

(0 1198791198672

(Ω)) 119906119905isin 1198712

(0 119879 1198712

(Ω))

119885 = 119871infin

(0 119879 1198712

(Ω))

(8)


intΩ

119908119867119870120601119889x

= intΩ

119867119905120601119889x + int

Ω

119863119867nabla119867nabla120601119889x + int

Ω

119889119867119867120601119889x

intΩ

119908119873119873(1 minus

119873

119870119873

minus 120579119870

119870119870

)120601119889x

= intΩ

119873119905120601119889x + int

Ω

119873119889119873119867120601119889x

intΩ

119908119870119870(1 minus

119870

119870119870

)120601119889x

= intΩ

119870119905120601119889x + int

Ω

119863119870(1 minus

119873

119870119873

)nabla119870nabla120601119889x

(9)


1198670isin 1198671

(Ω) cap 119862 (Ω) 1198730isin 119871infin

(Ω) cap 1198671

(Ω)

1198700isin 1198671

(Ω)

1198670ge 119862119867gt 0 0 lt 119873

0le119870119873

2 0 lt 119870

0le 119870119870

(10)


We set

119879 =

6

prod

119894=1

119879119894

(11)



(119867119898

119870119898

)119898isinN0


(119873119898

)119898isinN0

isin 119885

(12)



1205971198670

120597119905minus 119863119867Δ1198670

+ 1198891198671198670

= 0 (13)

1205971198730

120597119905+ 1198891198731198670

1198730

= 0 (14)

1205971198700

120597119905minus nabla sdot (119863

119870(1 minus

1198730

119870119873

)nabla1198700

) = 0 (15)

while (119867119898 119870119898)119898isinN0


isin 119885


120597119867119898+1

120597119905minus 119863119867Δ119867119898+1

+ 119889119867119867119898+1

= 119908119867119870119898

(16)

120597119873119898+1

120597119905+ 119889119873119867119898+1

119873119898+1

= 119908119873119873119898

(1 minus119873119898

119870119873

minus 120579119870119898

119870119870

)

(17)

120597119870119898+1

120597119905minus nabla sdot (119863

119870(1 minus

119873119898+1

119870119873

)nabla119870119898+1

)

= 119908119870119870119898

(1 minus119870119898

119870119870

)

(18)






0it holds that

119873119898

119873119898


((0 119879] times Ω) (19)

119867119898

119870119898

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

119867119898

119905 119870119898

119905isin 1198712

(0 119879 1198712

(Ω))

(20)




119867119898

(119905 x) ge 119862119867119890minus119889119867119905 119873

119898

(119905 x) le 119870119873

2

119870119898

(119905 x) le 119870119870



0the estimates

10038171003817100381710038171198671198981003817100381710038171003817119883

+100381710038171003817100381711986711989810038171003817100381710038171198712(01198791198672(Ω))

le 119862 (Ω 119879) (10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)+10038171003817100381710038171198670

10038171003817100381710038171198671(Ω))

(22)

10038171003817100381710038171198731198981003817100381710038171003817

2

119883le 119862 (Ω 119879)

100381710038171003817100381711987301003817100381710038171003817

2

1198671(Ω) (23)

10038171003817100381710038171198701198981003817100381710038171003817119883

+100381710038171003817100381711987011989810038171003817100381710038171198712(01198791198672(Ω))

le 2119862 (Ω 119879)10038171003817100381710038171198700

10038171003817100381710038171198671(Ω) (24)


119873119898

isin 119871infin

(0 119879 1198712

(Ω)) (25)





0

(119905 x) = 1198670 (119905 x) 119890119889119867119905 (26)


0

119905minus 119863119867Δ0

= 0 (27)


0isin 1198671(Ω) it


1198670

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

1198670

119905isin 1198712

(0 119879 1198712

(Ω))

(28)


+100381710038171003817100381710038171198670100381710038171003817100381710038171198712(01198791198672(Ω))

le 119862 (Ω 119879)10038171003817100381710038171198670

10038171003817100381710038171198671(Ω) (29)


1198730

(119905 x) = 1198730119890minusint

119905

01198891198731198670(119904x)119889119904

gt 0 (30)


10038171003817100381710038171003817119873010038171003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

=

10038171003817100381710038171003817100381710038171198730119890minusint

119905

01198891198731198670119889119905

1003817100381710038171003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

le10038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

(31)



show first that

1198730

isin 119871infin

((0 119879] times Ω) (32)

1198730


((0 119879] times Ω) (33)


(30)

=

10038171003817100381710038171003817100381710038171198730119890minusint

119905

01198891198731198670119889119905

1003817100381710038171003817100381710038171003817119871infin((0119879]timesΩ)

le10038171003817100381710038171198730

1003817100381710038171003817119871infin(Ω)lt infin

(34)

For 119905 ge 120575 gt 0 it is100381710038171003817100381710038171198730

119905

10038171003817100381710038171003817119871infin((0119879]timesΩ)

(30)

= 119889119873

10038171003817100381710038171003817100381710038171198730sdot 119890minusint

119905

01198891198731198670119889119905

sdot 1198670


(35)



0


Therefore


1198670

(119905 x) = lim(119905x)rarr (0x0)

0

(119905 x) 119890minus119889119867119905 = 1198670(x0)

(37)



119870(1 minus (119873

0(119905 x)119870

119873)) in (15)

depends on timeLet

119896119898(119905) =

119898

sum

119894=1

119889119894

119898(119905) 119908119894

(38)


119908119894infin


119908119894infin


(39)


119860 [119896119898 119896119898] = intΩ

119886 (119905 x) (nabla119896119898)2

119889x (40)



119889

119889119905119860 [119896119898 119896119898] = intΩ

1198861015840

(119905 x) (nabla119896119898)2

119889x

+ 2intΩ

119886 (119905 x) (nabla119896119898)1015840

nabla119896119898119889x

(41)


intΩ

1198861015840

(119905 x) (nabla119896119898)2


119863119870

119870119873

(1198730

)1015840

(119905 x) (nabla119896119898)2

119889x

(42)



100381710038171003817100381710038171198961015840

119898

10038171003817100381710038171003817

2

1198712(Ω)

+119889

119889119905(1

2119860 [119896119898 119896119898])

le119862

120577(1003817100381710038171003817119896119898

1003817100381710038171003817

2

1198671(Ω)

+10038171003817100381710038171198911003817100381710038171003817

2

1198712(Ω))

+ 2120577100381710038171003817100381710038171198961015840

119898

10038171003817100381710038171003817

2

1198712(Ω)

+1

2intΩ

119863119870

119870119873

(1198730

)1015840

(nabla119896119898)2

119889x

(43)


1198721198730 =

119863119870

119870119873

100381710038171003817100381710038171198730

119905

10038171003817100381710038171003817119871infin((0119879]timesΩ) (44)


int

119879

0

intΩ

119863119870

119870119873

(1198730

)1015840

(nabla119896119898)2

119889x 119889119905 le 1198721198730 int

119879

0

1003817100381710038171003817nabla1198961198981003817100381710038171003817

2

1198712(Ω)119889119905

le 1198721198730

10038171003817100381710038171198961198981003817100381710038171003817

2

1198712(0119879119867

1(Ω))

le 120574 (Ω 119879) lt infin

(45)




1198700

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

1198700

119905isin 1198712

(0 119879 1198712

(Ω))

10038171003817100381710038171003817119870010038171003817100381710038171003817119883

+100381710038171003817100381710038171198700100381710038171003817100381710038171198712(01198791198672(Ω))

le 119862 (Ω 119879)10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)

(46)

Since

1198700

0(x) gt 0 (47)





0


1= 1198881(Ω 119879)

int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198712(Ω)119889119905 le 119888

1int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)119889119905

ind hyple(24)

411988811198622

(Ω 119879) 11987910038171003817100381710038171198700

1003817100381710038171003817

2

1198671(Ω)

lt infin

(48)

and thus

119870119898

isin 1198712

(0 119879 1198712

(Ω)) (49)


119867119898+1

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

119867119898+1

119905isin 1198712

(0 119879 1198712

(Ω))

10038171003817100381710038171003817119867119898+110038171003817100381710038171003817119883

+10038171003817100381710038171003817119867119898+1100381710038171003817100381710038171198712(01198791198672(Ω))

le 1198621(Ω 119879) (2119908

119867119862 (Ω 119879)radic119888

111987910038171003817100381710038171198700

10038171003817100381710038171198671(Ω)+10038171003817100381710038171198670

10038171003817100381710038171198671(Ω))

le C (Ω 119879) (10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)+10038171003817100381710038171198670

10038171003817100381710038171198671(Ω))

(50)

withC(Ω 119879) = max1198621(Ω 119879) 119862

1(Ω 119879)2119908

119867119862(Ω 119879)radic119888

1119879


119898+1(119905 x) minus 119862

119867119890minus119889119867119905 for

which it holds

⟨120595119898+1

119905(119905) 120601⟩ + 119863

119867intΩ

nabla120595119898+1

nabla120601119889x + 119889119867intΩ

120595119898+1

120601119889x

= ⟨119908119867119870119898

120601⟩ (51)



119867119890minus119889119867119905


119873119898+1

(119905 x) = 119890minus120572(119905x) (1198730(x) + int

119905

0

120573 (119904 x) 119890120572(119904x)119889119904) (52)


where

120572 (119905 x) = int119905

0

119889119873119867119898+1

(V x) 119889V

120573 (119904 x) = 119908119873119873119898

(119904 x) (1 minus 119873119898

(119904 x)119870119873

minus 120579119870119898

(119904 x)119870119870

)

(53)


119873119898+1

isin 119871infin

((0 119879] times Ω) (54)

119873119898+1


((0 119879] times Ω) (55)


Next estimate10038171003817100381710038171003817119873119898+1

119905

10038171003817100381710038171003817119871infin((0119879]timesΩ)

le 119908119873

10038171003817100381710038171198731198981003817100381710038171003817119871infin((0119879]timesΩ)

10038171003817100381710038171003817100381710038171003817

1 minus119873119898

119870119873

minus 120579119870119898

119870119870

10038171003817100381710038171003817100381710038171003817119871infin((0119879]timesΩ)

+ 119889119873

10038171003817100381710038171003817119873119898+110038171003817100381710038171003817119871infin((0119879]timesΩ)

10038171003817100381710038171003817119867119898+110038171003817100381710038171003817119871infin((0119879]timesΩ)

lt infin

(56)



119871infin((0119879]timesΩ)


10038171003817100381710038171003817119873119898+1

119905




119873119898+1

(119905 x) le 119870119873

2119890minus120572(119905x)

+ int

119905

0

120573 (119904 x) 119890minus(120572(119905x)minus120572(119904x))119889119904

le119870119873

2119890minus120572(119905x)

+ 119908119873

119870119873

2int

119905

0

119890minus(120572(119905x)minus120572(119904x))

119889119904

(58)


119867119898+1


estimate

119873119898+1

(119905 x) le 119870119873

2119890minus119889119873119867119905 + 119908

119873

119870119873

2

1

119889119873119867

(1 minus 119890minus119889119873119867119905)

le119870119873

2((1 minus

119908119873

119889119873119867

) 119890minus119889119873119867119905 +

119908119873

119889119873119867

) le119870119873

2

(59)


Due to (52) we get

10038171003817100381710038171003817119873119898+1

(119905)10038171003817100381710038171003817

2

1198671(Ω)

=

10038171003817100381710038171003817100381710038171003817

119890minus120572(119905)

1198730+ 119890minus120572(119905)

int

119905

0

120573(119904)119890120572(119904)

119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 21199082

119873

10038171003817100381710038171003817100381710038171003817

int

119905

0

119873119898

(119904) (1 minus119873119898(119904)

119870119873

minus 120579119870119898(119904)

119870119870

)119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 21199082

119873

100381710038171003817100381710038171003817100381710038171003817

int

119905

0

(119873119898

(119904) minus(119873119898(119904))2

1198702

119873

)119889119904

100381710038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 41199082

119873(

10038171003817100381710038171003817100381710038171003817

int

119905

0

119873119898

(119904)119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

+1

1198702

119873

10038171003817100381710038171003817100381710038171003817

int

119905

0

(119873119898

(119904))2

119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

)

le10038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)[2 + 4119908

2

119873119862 (Ω 119879) 119879

2

]

le C (Ω 119879)10038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

(60)




int

119879

0

10038171003817100381710038171003817100381710038171003817

119870119898

(1 minus119870119898

119870119870

)

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le int

119879

0

(100381710038171003817100381711987011989810038171003817100381710038171198712(Ω)

+

1003817100381710038171003817100381710038171003817100381710038171003817

(119870119898)2

119870119870

10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

)

2

119889119905

le 2int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198712(Ω)119889119905 + 2int

119879

0

1003817100381710038171003817100381710038171003817100381710038171003817

(119870119898)2

119870119870

1003817100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 21198882

1int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)119889119905 + 2

1198884

2

1198702

119870

int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

4

1198671(Ω)119889119905

le 81198882

11198622

(Ω 119879)10038171003817100381710038171198700

1003817100381710038171003817

2

1198671(Ω)11987911198792

+ 321198884

2

1198702

119870

1198624

(Ω 119879)10038171003817100381710038171198700

1003817100381710038171003817

4

1198671(Ω)11987911198792lt infin

(61)


2(0 119879 119871

2(Ω)) and119870




119870119898+1

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

119870119898+1

119905isin 1198712

(0 119879 1198712

(Ω))

(62)


11198622(Ω 119879) 119879

11198624(Ω 119879) le 1

and

1198792= min1

2

1

161199082

1198701198882

1

100381710038171003817100381711987001003817100381710038171003817

1198702

119870

641199082

1198701198884

2

100381710038171003817100381711987001003817100381710038171003817

3 (63)

Then

int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905 le10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)(64)


+10038171003817100381710038171003817119870119898+1100381710038171003817100381710038171198712(01198791198672(Ω))

le 2119862 (Ω 119879)10038171003817100381710038171198700

10038171003817100381710038171198671(Ω) (65)


auxiliary function

120585119898+1


(119905 x) (66)


0

119870119898+1

le 119870119870 (67)





0


120597120585119898+1

120597119905minus nabla sdot (119863

119870(1 minus

119873119898+1

119870119873

)nabla120585119898+1

)


(1 minus119870119898

119870119870

)

(68)

Since

119870119898

(1 minus119870119898

119870119870

) le 119870119870 (69)



(1 minus119870119898

119870119870

) ge 0 (70)


3and 120578 such




(119905 x) = 120585119898+1 ge 0 (71)


119870119898+1

le 119870119870 (72)





0and





2(Ω) this



0 119867119898+1

0isin 1198671(Ω)


119867119898+1

0minus 119867119898

0isin 1198671

(Ω)

119870119898+1

minus 119870119898

isin 1198712

(0 119879 1198712

(Ω))

(73)



10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817119908119867119870119898

minus 119908119867119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

(74)


3= 1198883(Ω 119879) it follows that

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879)1199082

1198671198882

3int

119879

0

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le 119862 (Ω 119879)1199082

1198671198882

31198794

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

2

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(75)

where

1198794= min1

4

1

4119862 (Ω 119879)1199082

1198671198882

3

(76)




120597

120597119905(119873119898+1

minus 119873119898

) + 119889119873(119867119898+1

119873119898+1

minus 119867119898

119873119898

)

= 119908119873(119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

))

(77)


119898119870119873)minus(119870

119898119870119870))minus

119873119898minus1

(1 minus (119873119898minus1

119870119873) minus (119870

119898minus1119870119870)))




1

2intΩ

120597

120597119905(119873119898+1

minus 119873119898

)2

119889x

+ 119889119873intΩ

(119873119898+1

minus 119873119898

)2

119867119898+1

119889x

= intΩ

(ℎ (119873119898

119873119898minus1

) minus 119889119873119873119898

(119867119898+1

minus 119867119898

))

times (119873119898+1

minus 119873119898

) 119889x

(78)

Thus

119889

119889119905

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 2119908119873intΩ

10038161003816100381610038161003816100381610038161003816

(119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

))

times (119873119898+1

minus 119873119898

)

10038161003816100381610038161003816100381610038161003816

119889x

+ 2119889119873intΩ

10038161003816100381610038161003816119873119898

(119867119898+1

minus 119867119898

) (119873119898+1

minus 119873119898

)10038161003816100381610038161003816119889x

le [2119908119873

10038171003817100381710038171003817100381710038171003817

119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

)

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

+2119889119873

10038171003817100381710038171003817119873119898

(119867119898+1

minus 119867119898

)100381710038171003817100381710038171198712(Ω)

]

times10038171003817100381710038171003817119873119898+1

minus 119873119898100381710038171003817100381710038171198712(Ω)

(79)




5 times 107

cm3

119870119870

5 times 107

cm3

119908119873

1 times 10minus6

s119908119870

1 times 10minus6

s119863119870

2 times 10minus10

cm2s119863119867

5 times 10minus6

cm2s119908119867


119889119867

11 times 10minus4

s119889119873

0 rarr 10M sdot s

Let (recall (19))

119872max = max 119872119873119898 =

10038171003817100381710038171198731198981003817100381710038171003817119871infin((0119879]timesΩ)

119873119873119898minus1 =


(80)



2119908119873

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119873119898


minus(119873119898)2

119870119873

+

(119873119898minus1

)2

119870119873

minus119873119898119870119898

119870119870

+119873119898minus1

119870119898minus1

119870119870

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

le 2119908119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+4119908119873119872max119870119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 2119908119873119870119873

119870119870

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

+ 2119908119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

le 119862

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 119862

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

(81)

with 119862= 4119908

119873(1 + (119872max119870119873)) and 119862 = 2119908119873119870119873119870119870


2119889119873

10038171003817100381710038171003817119873119898

(119867119898+1

minus 119867119898

)100381710038171003817100381710038171198712(Ω)

le 1198891198731198884

119870119873

2

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

= 119862

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

(82)


119905 = 1

01

01

1

09

08

07

06

05

04

03

02

01090807060503 04020

119909

(a)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 10

119909

(b)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)



119889

119889119905

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le1

2(119862

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 119862

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

+119862

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

)

2

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

(83)


10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198901199052

int

119905

0

(1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

) 119889119904

(84)


01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 1

119909

(a)

01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 10

119909

(b)

01

09

08

07

06

05

04

03

02

001 1

1

090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)



1198622

we get

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le 119863 (Ω 119879) (10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

)1198795

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+5

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

)

(85)


119863 (Ω 119879) 1198795le1

4 (86)

Now since 1198701198980 119870119898+1

0isin 1198671(Ω) and 119870

119898(1 minus (119870

119898119870119870))

119870119898+1

(1 minus (119870119898+1

119870119870)) isin 119871

2(0 119879 119871

2(Ω)) we get

119870119898+1

0minus 119870119898

0isin 1198671

(Ω)

[119908119870119870119898+1

(1 minus119870119898+1

119870119870

) minus 119908119870119870119898

(1 minus119870119898

119870119870

)]

isin 1198712

(0 119879 1198712

(Ω))

(87)


055

05

045

04

035

03

025

02

015

01

0050 5 10 15 20 25 30 35 40

119905

119873

120575119873 = 50

120575119873 = 10

120575119873 = 2

(a)

119905 = 10

119873

07

06

05

04

03

02

01

00 01 02 03 04 05 06 07 09 108

120575119873 = 50

120575119873 = 10

120575119873 = 5

120575119873 = 2

120575119873 = 05

(b)



119905 = 1

1090807060504030201

00

051 0 02 04

08061

119910

119870

119909

(a)

119905 = 10

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

(b)

119905 = 50

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(c)

119905 = 1

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(d)

119905 = 10

109

08

0706

06

0504

04

0302

02

0100

1 0 02 040806

1

119910

119870

119909

(e)

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

119905 = 50

(f)



109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(a)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(b)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(c)

109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(d)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(e)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

minus119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

(88)


5= 1198885(Ω 119879) and 119888

6=


int

119879

0

100381710038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

) minus 119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1199082

119870

1198702

119870

100381710038171003817100381710038171003817(119870119898

)2


)2100381710038171003817100381710038171003817

2

1198712(Ω)

+ 1199082

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1198884

5

1199082

119870

1198702

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

10038171003817100381710038171003817119870119898

+ 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

+ 1199082

1198701198882

6

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le (int

119879

0

(41198884

5

1199082

119870

1198702

119870

[10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)

+10038171003817100381710038171003817119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

]

+21199082

1198701198882

6)119889119905)

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le (321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

1198796

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω)

+ 21199082

1198701198882

61198796)

times10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(89)

where

1198796= min 1

81

81205811

8120582

119896 = 321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω) 120582 = 2119908

2

1198701198882

6

(90)


109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(a)

060 02 04

08 106

002

04

04

081

119909

06

002

081

119910

119905 = 10

119873

(b)

109080706

06

0504030201

00

051 0 02 04 08 1

119909119910

119905 = 50

119873

(c)

109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(d)

109080706

06

0504030201

00

005 02 04 08 1119909119910

119905 = 10

119873

(e)

109080706

06

0504030201

100

051 0 02 04 08 1119909

119910

119905 = 50

119873

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le1

4(310038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

(91)


119871infin(0 119879 119871


solution follows


2 1198732 1198672) be two solu-


10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883 (92)

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883(93)

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le1

4

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+1

4

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883

+1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883

(94)


1= 1198672


of the uniqueness








119870 (0 119909) = exp(minus|119909|2

120598) 120598 gt 0


120598)) 120598 gt 0

119867 (0 119909) = 120577119870 (0 119909) 120577 isin [0 1)

(95)




119894 where 119894 = 1 2 119898 + 1 and 119894 = 1 2 (119898 + 1)

2


119867119899+1

119894minus 119867119899

119894

Δ119905= 120575119867119870119899

119894minus 120575119867119867119899+1

119894+ (

119867119899+1

119894minus1minus 2119867119899+1

119894+ 119867119899+1

119894+1

Δ1199092)

(96)


AHH119899+1

= H119899 + 120599119899H (97)




119873119899+1

119894=

1

1 + Δ119905120575119873119867119899+1

119894

[119873119899

119894+ Δ119905119873

119899

119894(1 minus 120579119870

119899

119894minus 119873119899

119894)] (98)



nabla (119863 (119873)nabla119870)|119909119894=

1

2(Δ119909)2sum

119896isin119873119894

(119863 (119873119899+1

119896) + 119863 (119873

119899+1

119894))

times (119870119899+1

119896minus 119870119899+1

119894)

(99)

where119863(119873) = 120575119896(1 minus119873) and119873



119894has two


AKK119899+1

= K119899 + 120599K (100)


119896119870119899

119894(1minus

119870119899

119894minus 119873119899+1


























119867and larger 120575













119873=





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119867119898

(119905 x) ge 119862119867119890minus119889119867119905 119873

119898

(119905 x) le 119870119873

2

119870119898

(119905 x) le 119870119870



0the estimates

10038171003817100381710038171198671198981003817100381710038171003817119883

+100381710038171003817100381711986711989810038171003817100381710038171198712(01198791198672(Ω))

le 119862 (Ω 119879) (10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)+10038171003817100381710038171198670

10038171003817100381710038171198671(Ω))

(22)

10038171003817100381710038171198731198981003817100381710038171003817

2

119883le 119862 (Ω 119879)

100381710038171003817100381711987301003817100381710038171003817

2

1198671(Ω) (23)

10038171003817100381710038171198701198981003817100381710038171003817119883

+100381710038171003817100381711987011989810038171003817100381710038171198712(01198791198672(Ω))

le 2119862 (Ω 119879)10038171003817100381710038171198700

10038171003817100381710038171198671(Ω) (24)


119873119898

isin 119871infin

(0 119879 1198712

(Ω)) (25)





0

(119905 x) = 1198670 (119905 x) 119890119889119867119905 (26)


0

119905minus 119863119867Δ0

= 0 (27)


0isin 1198671(Ω) it


1198670

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

1198670

119905isin 1198712

(0 119879 1198712

(Ω))

(28)


+100381710038171003817100381710038171198670100381710038171003817100381710038171198712(01198791198672(Ω))

le 119862 (Ω 119879)10038171003817100381710038171198670

10038171003817100381710038171198671(Ω) (29)


1198730

(119905 x) = 1198730119890minusint

119905

01198891198731198670(119904x)119889119904

gt 0 (30)


10038171003817100381710038171003817119873010038171003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

=

10038171003817100381710038171003817100381710038171198730119890minusint

119905

01198891198731198670119889119905

1003817100381710038171003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

le10038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

(31)



show first that

1198730

isin 119871infin

((0 119879] times Ω) (32)

1198730


((0 119879] times Ω) (33)


(30)

=

10038171003817100381710038171003817100381710038171198730119890minusint

119905

01198891198731198670119889119905

1003817100381710038171003817100381710038171003817119871infin((0119879]timesΩ)

le10038171003817100381710038171198730

1003817100381710038171003817119871infin(Ω)lt infin

(34)

For 119905 ge 120575 gt 0 it is100381710038171003817100381710038171198730

119905

10038171003817100381710038171003817119871infin((0119879]timesΩ)

(30)

= 119889119873

10038171003817100381710038171003817100381710038171198730sdot 119890minusint

119905

01198891198731198670119889119905

sdot 1198670


(35)



0


Therefore


1198670

(119905 x) = lim(119905x)rarr (0x0)

0

(119905 x) 119890minus119889119867119905 = 1198670(x0)

(37)



119870(1 minus (119873

0(119905 x)119870

119873)) in (15)

depends on timeLet

119896119898(119905) =

119898

sum

119894=1

119889119894

119898(119905) 119908119894

(38)


119908119894infin


119908119894infin


(39)


119860 [119896119898 119896119898] = intΩ

119886 (119905 x) (nabla119896119898)2

119889x (40)



119889

119889119905119860 [119896119898 119896119898] = intΩ

1198861015840

(119905 x) (nabla119896119898)2

119889x

+ 2intΩ

119886 (119905 x) (nabla119896119898)1015840

nabla119896119898119889x

(41)


intΩ

1198861015840

(119905 x) (nabla119896119898)2


119863119870

119870119873

(1198730

)1015840

(119905 x) (nabla119896119898)2

119889x

(42)



100381710038171003817100381710038171198961015840

119898

10038171003817100381710038171003817

2

1198712(Ω)

+119889

119889119905(1

2119860 [119896119898 119896119898])

le119862

120577(1003817100381710038171003817119896119898

1003817100381710038171003817

2

1198671(Ω)

+10038171003817100381710038171198911003817100381710038171003817

2

1198712(Ω))

+ 2120577100381710038171003817100381710038171198961015840

119898

10038171003817100381710038171003817

2

1198712(Ω)

+1

2intΩ

119863119870

119870119873

(1198730

)1015840

(nabla119896119898)2

119889x

(43)


1198721198730 =

119863119870

119870119873

100381710038171003817100381710038171198730

119905

10038171003817100381710038171003817119871infin((0119879]timesΩ) (44)


int

119879

0

intΩ

119863119870

119870119873

(1198730

)1015840

(nabla119896119898)2

119889x 119889119905 le 1198721198730 int

119879

0

1003817100381710038171003817nabla1198961198981003817100381710038171003817

2

1198712(Ω)119889119905

le 1198721198730

10038171003817100381710038171198961198981003817100381710038171003817

2

1198712(0119879119867

1(Ω))

le 120574 (Ω 119879) lt infin

(45)




1198700

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

1198700

119905isin 1198712

(0 119879 1198712

(Ω))

10038171003817100381710038171003817119870010038171003817100381710038171003817119883

+100381710038171003817100381710038171198700100381710038171003817100381710038171198712(01198791198672(Ω))

le 119862 (Ω 119879)10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)

(46)

Since

1198700

0(x) gt 0 (47)





0


1= 1198881(Ω 119879)

int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198712(Ω)119889119905 le 119888

1int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)119889119905

ind hyple(24)

411988811198622

(Ω 119879) 11987910038171003817100381710038171198700

1003817100381710038171003817

2

1198671(Ω)

lt infin

(48)

and thus

119870119898

isin 1198712

(0 119879 1198712

(Ω)) (49)


119867119898+1

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

119867119898+1

119905isin 1198712

(0 119879 1198712

(Ω))

10038171003817100381710038171003817119867119898+110038171003817100381710038171003817119883

+10038171003817100381710038171003817119867119898+1100381710038171003817100381710038171198712(01198791198672(Ω))

le 1198621(Ω 119879) (2119908

119867119862 (Ω 119879)radic119888

111987910038171003817100381710038171198700

10038171003817100381710038171198671(Ω)+10038171003817100381710038171198670

10038171003817100381710038171198671(Ω))

le C (Ω 119879) (10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)+10038171003817100381710038171198670

10038171003817100381710038171198671(Ω))

(50)

withC(Ω 119879) = max1198621(Ω 119879) 119862

1(Ω 119879)2119908

119867119862(Ω 119879)radic119888

1119879


119898+1(119905 x) minus 119862

119867119890minus119889119867119905 for

which it holds

⟨120595119898+1

119905(119905) 120601⟩ + 119863

119867intΩ

nabla120595119898+1

nabla120601119889x + 119889119867intΩ

120595119898+1

120601119889x

= ⟨119908119867119870119898

120601⟩ (51)



119867119890minus119889119867119905


119873119898+1

(119905 x) = 119890minus120572(119905x) (1198730(x) + int

119905

0

120573 (119904 x) 119890120572(119904x)119889119904) (52)


where

120572 (119905 x) = int119905

0

119889119873119867119898+1

(V x) 119889V

120573 (119904 x) = 119908119873119873119898

(119904 x) (1 minus 119873119898

(119904 x)119870119873

minus 120579119870119898

(119904 x)119870119870

)

(53)


119873119898+1

isin 119871infin

((0 119879] times Ω) (54)

119873119898+1


((0 119879] times Ω) (55)


Next estimate10038171003817100381710038171003817119873119898+1

119905

10038171003817100381710038171003817119871infin((0119879]timesΩ)

le 119908119873

10038171003817100381710038171198731198981003817100381710038171003817119871infin((0119879]timesΩ)

10038171003817100381710038171003817100381710038171003817

1 minus119873119898

119870119873

minus 120579119870119898

119870119870

10038171003817100381710038171003817100381710038171003817119871infin((0119879]timesΩ)

+ 119889119873

10038171003817100381710038171003817119873119898+110038171003817100381710038171003817119871infin((0119879]timesΩ)

10038171003817100381710038171003817119867119898+110038171003817100381710038171003817119871infin((0119879]timesΩ)

lt infin

(56)



119871infin((0119879]timesΩ)


10038171003817100381710038171003817119873119898+1

119905




119873119898+1

(119905 x) le 119870119873

2119890minus120572(119905x)

+ int

119905

0

120573 (119904 x) 119890minus(120572(119905x)minus120572(119904x))119889119904

le119870119873

2119890minus120572(119905x)

+ 119908119873

119870119873

2int

119905

0

119890minus(120572(119905x)minus120572(119904x))

119889119904

(58)


119867119898+1


estimate

119873119898+1

(119905 x) le 119870119873

2119890minus119889119873119867119905 + 119908

119873

119870119873

2

1

119889119873119867

(1 minus 119890minus119889119873119867119905)

le119870119873

2((1 minus

119908119873

119889119873119867

) 119890minus119889119873119867119905 +

119908119873

119889119873119867

) le119870119873

2

(59)


Due to (52) we get

10038171003817100381710038171003817119873119898+1

(119905)10038171003817100381710038171003817

2

1198671(Ω)

=

10038171003817100381710038171003817100381710038171003817

119890minus120572(119905)

1198730+ 119890minus120572(119905)

int

119905

0

120573(119904)119890120572(119904)

119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 21199082

119873

10038171003817100381710038171003817100381710038171003817

int

119905

0

119873119898

(119904) (1 minus119873119898(119904)

119870119873

minus 120579119870119898(119904)

119870119870

)119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 21199082

119873

100381710038171003817100381710038171003817100381710038171003817

int

119905

0

(119873119898

(119904) minus(119873119898(119904))2

1198702

119873

)119889119904

100381710038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 41199082

119873(

10038171003817100381710038171003817100381710038171003817

int

119905

0

119873119898

(119904)119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

+1

1198702

119873

10038171003817100381710038171003817100381710038171003817

int

119905

0

(119873119898

(119904))2

119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

)

le10038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)[2 + 4119908

2

119873119862 (Ω 119879) 119879

2

]

le C (Ω 119879)10038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

(60)




int

119879

0

10038171003817100381710038171003817100381710038171003817

119870119898

(1 minus119870119898

119870119870

)

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le int

119879

0

(100381710038171003817100381711987011989810038171003817100381710038171198712(Ω)

+

1003817100381710038171003817100381710038171003817100381710038171003817

(119870119898)2

119870119870

10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

)

2

119889119905

le 2int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198712(Ω)119889119905 + 2int

119879

0

1003817100381710038171003817100381710038171003817100381710038171003817

(119870119898)2

119870119870

1003817100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 21198882

1int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)119889119905 + 2

1198884

2

1198702

119870

int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

4

1198671(Ω)119889119905

le 81198882

11198622

(Ω 119879)10038171003817100381710038171198700

1003817100381710038171003817

2

1198671(Ω)11987911198792

+ 321198884

2

1198702

119870

1198624

(Ω 119879)10038171003817100381710038171198700

1003817100381710038171003817

4

1198671(Ω)11987911198792lt infin

(61)


2(0 119879 119871

2(Ω)) and119870




119870119898+1

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

119870119898+1

119905isin 1198712

(0 119879 1198712

(Ω))

(62)


11198622(Ω 119879) 119879

11198624(Ω 119879) le 1

and

1198792= min1

2

1

161199082

1198701198882

1

100381710038171003817100381711987001003817100381710038171003817

1198702

119870

641199082

1198701198884

2

100381710038171003817100381711987001003817100381710038171003817

3 (63)

Then

int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905 le10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)(64)


+10038171003817100381710038171003817119870119898+1100381710038171003817100381710038171198712(01198791198672(Ω))

le 2119862 (Ω 119879)10038171003817100381710038171198700

10038171003817100381710038171198671(Ω) (65)


auxiliary function

120585119898+1


(119905 x) (66)


0

119870119898+1

le 119870119870 (67)





0


120597120585119898+1

120597119905minus nabla sdot (119863

119870(1 minus

119873119898+1

119870119873

)nabla120585119898+1

)


(1 minus119870119898

119870119870

)

(68)

Since

119870119898

(1 minus119870119898

119870119870

) le 119870119870 (69)



(1 minus119870119898

119870119870

) ge 0 (70)


3and 120578 such




(119905 x) = 120585119898+1 ge 0 (71)


119870119898+1

le 119870119870 (72)





0and





2(Ω) this



0 119867119898+1

0isin 1198671(Ω)


119867119898+1

0minus 119867119898

0isin 1198671

(Ω)

119870119898+1

minus 119870119898

isin 1198712

(0 119879 1198712

(Ω))

(73)



10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817119908119867119870119898

minus 119908119867119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

(74)


3= 1198883(Ω 119879) it follows that

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879)1199082

1198671198882

3int

119879

0

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le 119862 (Ω 119879)1199082

1198671198882

31198794

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

2

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(75)

where

1198794= min1

4

1

4119862 (Ω 119879)1199082

1198671198882

3

(76)




120597

120597119905(119873119898+1

minus 119873119898

) + 119889119873(119867119898+1

119873119898+1

minus 119867119898

119873119898

)

= 119908119873(119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

))

(77)


119898119870119873)minus(119870

119898119870119870))minus

119873119898minus1

(1 minus (119873119898minus1

119870119873) minus (119870

119898minus1119870119870)))




1

2intΩ

120597

120597119905(119873119898+1

minus 119873119898

)2

119889x

+ 119889119873intΩ

(119873119898+1

minus 119873119898

)2

119867119898+1

119889x

= intΩ

(ℎ (119873119898

119873119898minus1

) minus 119889119873119873119898

(119867119898+1

minus 119867119898

))

times (119873119898+1

minus 119873119898

) 119889x

(78)

Thus

119889

119889119905

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 2119908119873intΩ

10038161003816100381610038161003816100381610038161003816

(119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

))

times (119873119898+1

minus 119873119898

)

10038161003816100381610038161003816100381610038161003816

119889x

+ 2119889119873intΩ

10038161003816100381610038161003816119873119898

(119867119898+1

minus 119867119898

) (119873119898+1

minus 119873119898

)10038161003816100381610038161003816119889x

le [2119908119873

10038171003817100381710038171003817100381710038171003817

119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

)

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

+2119889119873

10038171003817100381710038171003817119873119898

(119867119898+1

minus 119867119898

)100381710038171003817100381710038171198712(Ω)

]

times10038171003817100381710038171003817119873119898+1

minus 119873119898100381710038171003817100381710038171198712(Ω)

(79)




5 times 107

cm3

119870119870

5 times 107

cm3

119908119873

1 times 10minus6

s119908119870

1 times 10minus6

s119863119870

2 times 10minus10

cm2s119863119867

5 times 10minus6

cm2s119908119867


119889119867

11 times 10minus4

s119889119873

0 rarr 10M sdot s

Let (recall (19))

119872max = max 119872119873119898 =

10038171003817100381710038171198731198981003817100381710038171003817119871infin((0119879]timesΩ)

119873119873119898minus1 =


(80)



2119908119873

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119873119898


minus(119873119898)2

119870119873

+

(119873119898minus1

)2

119870119873

minus119873119898119870119898

119870119870

+119873119898minus1

119870119898minus1

119870119870

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

le 2119908119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+4119908119873119872max119870119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 2119908119873119870119873

119870119870

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

+ 2119908119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

le 119862

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 119862

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

(81)

with 119862= 4119908

119873(1 + (119872max119870119873)) and 119862 = 2119908119873119870119873119870119870


2119889119873

10038171003817100381710038171003817119873119898

(119867119898+1

minus 119867119898

)100381710038171003817100381710038171198712(Ω)

le 1198891198731198884

119870119873

2

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

= 119862

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

(82)


119905 = 1

01

01

1

09

08

07

06

05

04

03

02

01090807060503 04020

119909

(a)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 10

119909

(b)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)



119889

119889119905

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le1

2(119862

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 119862

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

+119862

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

)

2

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

(83)


10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198901199052

int

119905

0

(1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

) 119889119904

(84)


01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 1

119909

(a)

01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 10

119909

(b)

01

09

08

07

06

05

04

03

02

001 1

1

090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)



1198622

we get

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le 119863 (Ω 119879) (10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

)1198795

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+5

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

)

(85)


119863 (Ω 119879) 1198795le1

4 (86)

Now since 1198701198980 119870119898+1

0isin 1198671(Ω) and 119870

119898(1 minus (119870

119898119870119870))

119870119898+1

(1 minus (119870119898+1

119870119870)) isin 119871

2(0 119879 119871

2(Ω)) we get

119870119898+1

0minus 119870119898

0isin 1198671

(Ω)

[119908119870119870119898+1

(1 minus119870119898+1

119870119870

) minus 119908119870119870119898

(1 minus119870119898

119870119870

)]

isin 1198712

(0 119879 1198712

(Ω))

(87)


055

05

045

04

035

03

025

02

015

01

0050 5 10 15 20 25 30 35 40

119905

119873

120575119873 = 50

120575119873 = 10

120575119873 = 2

(a)

119905 = 10

119873

07

06

05

04

03

02

01

00 01 02 03 04 05 06 07 09 108

120575119873 = 50

120575119873 = 10

120575119873 = 5

120575119873 = 2

120575119873 = 05

(b)



119905 = 1

1090807060504030201

00

051 0 02 04

08061

119910

119870

119909

(a)

119905 = 10

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

(b)

119905 = 50

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(c)

119905 = 1

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(d)

119905 = 10

109

08

0706

06

0504

04

0302

02

0100

1 0 02 040806

1

119910

119870

119909

(e)

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

119905 = 50

(f)



109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(a)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(b)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(c)

109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(d)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(e)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

minus119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

(88)


5= 1198885(Ω 119879) and 119888

6=


int

119879

0

100381710038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

) minus 119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1199082

119870

1198702

119870

100381710038171003817100381710038171003817(119870119898

)2


)2100381710038171003817100381710038171003817

2

1198712(Ω)

+ 1199082

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1198884

5

1199082

119870

1198702

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

10038171003817100381710038171003817119870119898

+ 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

+ 1199082

1198701198882

6

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le (int

119879

0

(41198884

5

1199082

119870

1198702

119870

[10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)

+10038171003817100381710038171003817119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

]

+21199082

1198701198882

6)119889119905)

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le (321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

1198796

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω)

+ 21199082

1198701198882

61198796)

times10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(89)

where

1198796= min 1

81

81205811

8120582

119896 = 321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω) 120582 = 2119908

2

1198701198882

6

(90)


109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(a)

060 02 04

08 106

002

04

04

081

119909

06

002

081

119910

119905 = 10

119873

(b)

109080706

06

0504030201

00

051 0 02 04 08 1

119909119910

119905 = 50

119873

(c)

109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(d)

109080706

06

0504030201

00

005 02 04 08 1119909119910

119905 = 10

119873

(e)

109080706

06

0504030201

100

051 0 02 04 08 1119909

119910

119905 = 50

119873

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le1

4(310038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

(91)


119871infin(0 119879 119871


solution follows


2 1198732 1198672) be two solu-


10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883 (92)

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883(93)

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le1

4

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+1

4

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883

+1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883

(94)


1= 1198672


of the uniqueness








119870 (0 119909) = exp(minus|119909|2

120598) 120598 gt 0


120598)) 120598 gt 0

119867 (0 119909) = 120577119870 (0 119909) 120577 isin [0 1)

(95)




119894 where 119894 = 1 2 119898 + 1 and 119894 = 1 2 (119898 + 1)

2


119867119899+1

119894minus 119867119899

119894

Δ119905= 120575119867119870119899

119894minus 120575119867119867119899+1

119894+ (

119867119899+1

119894minus1minus 2119867119899+1

119894+ 119867119899+1

119894+1

Δ1199092)

(96)


AHH119899+1

= H119899 + 120599119899H (97)




119873119899+1

119894=

1

1 + Δ119905120575119873119867119899+1

119894

[119873119899

119894+ Δ119905119873

119899

119894(1 minus 120579119870

119899

119894minus 119873119899

119894)] (98)



nabla (119863 (119873)nabla119870)|119909119894=

1

2(Δ119909)2sum

119896isin119873119894

(119863 (119873119899+1

119896) + 119863 (119873

119899+1

119894))

times (119870119899+1

119896minus 119870119899+1

119894)

(99)

where119863(119873) = 120575119896(1 minus119873) and119873



119894has two


AKK119899+1

= K119899 + 120599K (100)


119896119870119899

119894(1minus

119870119899

119894minus 119873119899+1


























119867and larger 120575













119873=





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119889

119889119905119860 [119896119898 119896119898] = intΩ

1198861015840

(119905 x) (nabla119896119898)2

119889x

+ 2intΩ

119886 (119905 x) (nabla119896119898)1015840

nabla119896119898119889x

(41)


intΩ

1198861015840

(119905 x) (nabla119896119898)2


119863119870

119870119873

(1198730

)1015840

(119905 x) (nabla119896119898)2

119889x

(42)



100381710038171003817100381710038171198961015840

119898

10038171003817100381710038171003817

2

1198712(Ω)

+119889

119889119905(1

2119860 [119896119898 119896119898])

le119862

120577(1003817100381710038171003817119896119898

1003817100381710038171003817

2

1198671(Ω)

+10038171003817100381710038171198911003817100381710038171003817

2

1198712(Ω))

+ 2120577100381710038171003817100381710038171198961015840

119898

10038171003817100381710038171003817

2

1198712(Ω)

+1

2intΩ

119863119870

119870119873

(1198730

)1015840

(nabla119896119898)2

119889x

(43)


1198721198730 =

119863119870

119870119873

100381710038171003817100381710038171198730

119905

10038171003817100381710038171003817119871infin((0119879]timesΩ) (44)


int

119879

0

intΩ

119863119870

119870119873

(1198730

)1015840

(nabla119896119898)2

119889x 119889119905 le 1198721198730 int

119879

0

1003817100381710038171003817nabla1198961198981003817100381710038171003817

2

1198712(Ω)119889119905

le 1198721198730

10038171003817100381710038171198961198981003817100381710038171003817

2

1198712(0119879119867

1(Ω))

le 120574 (Ω 119879) lt infin

(45)




1198700

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

1198700

119905isin 1198712

(0 119879 1198712

(Ω))

10038171003817100381710038171003817119870010038171003817100381710038171003817119883

+100381710038171003817100381710038171198700100381710038171003817100381710038171198712(01198791198672(Ω))

le 119862 (Ω 119879)10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)

(46)

Since

1198700

0(x) gt 0 (47)





0


1= 1198881(Ω 119879)

int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198712(Ω)119889119905 le 119888

1int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)119889119905

ind hyple(24)

411988811198622

(Ω 119879) 11987910038171003817100381710038171198700

1003817100381710038171003817

2

1198671(Ω)

lt infin

(48)

and thus

119870119898

isin 1198712

(0 119879 1198712

(Ω)) (49)


119867119898+1

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

119867119898+1

119905isin 1198712

(0 119879 1198712

(Ω))

10038171003817100381710038171003817119867119898+110038171003817100381710038171003817119883

+10038171003817100381710038171003817119867119898+1100381710038171003817100381710038171198712(01198791198672(Ω))

le 1198621(Ω 119879) (2119908

119867119862 (Ω 119879)radic119888

111987910038171003817100381710038171198700

10038171003817100381710038171198671(Ω)+10038171003817100381710038171198670

10038171003817100381710038171198671(Ω))

le C (Ω 119879) (10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)+10038171003817100381710038171198670

10038171003817100381710038171198671(Ω))

(50)

withC(Ω 119879) = max1198621(Ω 119879) 119862

1(Ω 119879)2119908

119867119862(Ω 119879)radic119888

1119879


119898+1(119905 x) minus 119862

119867119890minus119889119867119905 for

which it holds

⟨120595119898+1

119905(119905) 120601⟩ + 119863

119867intΩ

nabla120595119898+1

nabla120601119889x + 119889119867intΩ

120595119898+1

120601119889x

= ⟨119908119867119870119898

120601⟩ (51)



119867119890minus119889119867119905


119873119898+1

(119905 x) = 119890minus120572(119905x) (1198730(x) + int

119905

0

120573 (119904 x) 119890120572(119904x)119889119904) (52)


where

120572 (119905 x) = int119905

0

119889119873119867119898+1

(V x) 119889V

120573 (119904 x) = 119908119873119873119898

(119904 x) (1 minus 119873119898

(119904 x)119870119873

minus 120579119870119898

(119904 x)119870119870

)

(53)


119873119898+1

isin 119871infin

((0 119879] times Ω) (54)

119873119898+1


((0 119879] times Ω) (55)


Next estimate10038171003817100381710038171003817119873119898+1

119905

10038171003817100381710038171003817119871infin((0119879]timesΩ)

le 119908119873

10038171003817100381710038171198731198981003817100381710038171003817119871infin((0119879]timesΩ)

10038171003817100381710038171003817100381710038171003817

1 minus119873119898

119870119873

minus 120579119870119898

119870119870

10038171003817100381710038171003817100381710038171003817119871infin((0119879]timesΩ)

+ 119889119873

10038171003817100381710038171003817119873119898+110038171003817100381710038171003817119871infin((0119879]timesΩ)

10038171003817100381710038171003817119867119898+110038171003817100381710038171003817119871infin((0119879]timesΩ)

lt infin

(56)



119871infin((0119879]timesΩ)


10038171003817100381710038171003817119873119898+1

119905




119873119898+1

(119905 x) le 119870119873

2119890minus120572(119905x)

+ int

119905

0

120573 (119904 x) 119890minus(120572(119905x)minus120572(119904x))119889119904

le119870119873

2119890minus120572(119905x)

+ 119908119873

119870119873

2int

119905

0

119890minus(120572(119905x)minus120572(119904x))

119889119904

(58)


119867119898+1


estimate

119873119898+1

(119905 x) le 119870119873

2119890minus119889119873119867119905 + 119908

119873

119870119873

2

1

119889119873119867

(1 minus 119890minus119889119873119867119905)

le119870119873

2((1 minus

119908119873

119889119873119867

) 119890minus119889119873119867119905 +

119908119873

119889119873119867

) le119870119873

2

(59)


Due to (52) we get

10038171003817100381710038171003817119873119898+1

(119905)10038171003817100381710038171003817

2

1198671(Ω)

=

10038171003817100381710038171003817100381710038171003817

119890minus120572(119905)

1198730+ 119890minus120572(119905)

int

119905

0

120573(119904)119890120572(119904)

119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 21199082

119873

10038171003817100381710038171003817100381710038171003817

int

119905

0

119873119898

(119904) (1 minus119873119898(119904)

119870119873

minus 120579119870119898(119904)

119870119870

)119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 21199082

119873

100381710038171003817100381710038171003817100381710038171003817

int

119905

0

(119873119898

(119904) minus(119873119898(119904))2

1198702

119873

)119889119904

100381710038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 41199082

119873(

10038171003817100381710038171003817100381710038171003817

int

119905

0

119873119898

(119904)119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

+1

1198702

119873

10038171003817100381710038171003817100381710038171003817

int

119905

0

(119873119898

(119904))2

119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

)

le10038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)[2 + 4119908

2

119873119862 (Ω 119879) 119879

2

]

le C (Ω 119879)10038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

(60)




int

119879

0

10038171003817100381710038171003817100381710038171003817

119870119898

(1 minus119870119898

119870119870

)

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le int

119879

0

(100381710038171003817100381711987011989810038171003817100381710038171198712(Ω)

+

1003817100381710038171003817100381710038171003817100381710038171003817

(119870119898)2

119870119870

10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

)

2

119889119905

le 2int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198712(Ω)119889119905 + 2int

119879

0

1003817100381710038171003817100381710038171003817100381710038171003817

(119870119898)2

119870119870

1003817100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 21198882

1int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)119889119905 + 2

1198884

2

1198702

119870

int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

4

1198671(Ω)119889119905

le 81198882

11198622

(Ω 119879)10038171003817100381710038171198700

1003817100381710038171003817

2

1198671(Ω)11987911198792

+ 321198884

2

1198702

119870

1198624

(Ω 119879)10038171003817100381710038171198700

1003817100381710038171003817

4

1198671(Ω)11987911198792lt infin

(61)


2(0 119879 119871

2(Ω)) and119870




119870119898+1

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

119870119898+1

119905isin 1198712

(0 119879 1198712

(Ω))

(62)


11198622(Ω 119879) 119879

11198624(Ω 119879) le 1

and

1198792= min1

2

1

161199082

1198701198882

1

100381710038171003817100381711987001003817100381710038171003817

1198702

119870

641199082

1198701198884

2

100381710038171003817100381711987001003817100381710038171003817

3 (63)

Then

int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905 le10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)(64)


+10038171003817100381710038171003817119870119898+1100381710038171003817100381710038171198712(01198791198672(Ω))

le 2119862 (Ω 119879)10038171003817100381710038171198700

10038171003817100381710038171198671(Ω) (65)


auxiliary function

120585119898+1


(119905 x) (66)


0

119870119898+1

le 119870119870 (67)





0


120597120585119898+1

120597119905minus nabla sdot (119863

119870(1 minus

119873119898+1

119870119873

)nabla120585119898+1

)


(1 minus119870119898

119870119870

)

(68)

Since

119870119898

(1 minus119870119898

119870119870

) le 119870119870 (69)



(1 minus119870119898

119870119870

) ge 0 (70)


3and 120578 such




(119905 x) = 120585119898+1 ge 0 (71)


119870119898+1

le 119870119870 (72)





0and





2(Ω) this



0 119867119898+1

0isin 1198671(Ω)


119867119898+1

0minus 119867119898

0isin 1198671

(Ω)

119870119898+1

minus 119870119898

isin 1198712

(0 119879 1198712

(Ω))

(73)



10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817119908119867119870119898

minus 119908119867119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

(74)


3= 1198883(Ω 119879) it follows that

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879)1199082

1198671198882

3int

119879

0

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le 119862 (Ω 119879)1199082

1198671198882

31198794

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

2

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(75)

where

1198794= min1

4

1

4119862 (Ω 119879)1199082

1198671198882

3

(76)




120597

120597119905(119873119898+1

minus 119873119898

) + 119889119873(119867119898+1

119873119898+1

minus 119867119898

119873119898

)

= 119908119873(119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

))

(77)


119898119870119873)minus(119870

119898119870119870))minus

119873119898minus1

(1 minus (119873119898minus1

119870119873) minus (119870

119898minus1119870119870)))




1

2intΩ

120597

120597119905(119873119898+1

minus 119873119898

)2

119889x

+ 119889119873intΩ

(119873119898+1

minus 119873119898

)2

119867119898+1

119889x

= intΩ

(ℎ (119873119898

119873119898minus1

) minus 119889119873119873119898

(119867119898+1

minus 119867119898

))

times (119873119898+1

minus 119873119898

) 119889x

(78)

Thus

119889

119889119905

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 2119908119873intΩ

10038161003816100381610038161003816100381610038161003816

(119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

))

times (119873119898+1

minus 119873119898

)

10038161003816100381610038161003816100381610038161003816

119889x

+ 2119889119873intΩ

10038161003816100381610038161003816119873119898

(119867119898+1

minus 119867119898

) (119873119898+1

minus 119873119898

)10038161003816100381610038161003816119889x

le [2119908119873

10038171003817100381710038171003817100381710038171003817

119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

)

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

+2119889119873

10038171003817100381710038171003817119873119898

(119867119898+1

minus 119867119898

)100381710038171003817100381710038171198712(Ω)

]

times10038171003817100381710038171003817119873119898+1

minus 119873119898100381710038171003817100381710038171198712(Ω)

(79)




5 times 107

cm3

119870119870

5 times 107

cm3

119908119873

1 times 10minus6

s119908119870

1 times 10minus6

s119863119870

2 times 10minus10

cm2s119863119867

5 times 10minus6

cm2s119908119867


119889119867

11 times 10minus4

s119889119873

0 rarr 10M sdot s

Let (recall (19))

119872max = max 119872119873119898 =

10038171003817100381710038171198731198981003817100381710038171003817119871infin((0119879]timesΩ)

119873119873119898minus1 =


(80)



2119908119873

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119873119898


minus(119873119898)2

119870119873

+

(119873119898minus1

)2

119870119873

minus119873119898119870119898

119870119870

+119873119898minus1

119870119898minus1

119870119870

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

le 2119908119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+4119908119873119872max119870119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 2119908119873119870119873

119870119870

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

+ 2119908119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

le 119862

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 119862

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

(81)

with 119862= 4119908

119873(1 + (119872max119870119873)) and 119862 = 2119908119873119870119873119870119870


2119889119873

10038171003817100381710038171003817119873119898

(119867119898+1

minus 119867119898

)100381710038171003817100381710038171198712(Ω)

le 1198891198731198884

119870119873

2

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

= 119862

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

(82)


119905 = 1

01

01

1

09

08

07

06

05

04

03

02

01090807060503 04020

119909

(a)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 10

119909

(b)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)



119889

119889119905

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le1

2(119862

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 119862

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

+119862

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

)

2

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

(83)


10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198901199052

int

119905

0

(1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

) 119889119904

(84)


01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 1

119909

(a)

01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 10

119909

(b)

01

09

08

07

06

05

04

03

02

001 1

1

090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)



1198622

we get

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le 119863 (Ω 119879) (10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

)1198795

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+5

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

)

(85)


119863 (Ω 119879) 1198795le1

4 (86)

Now since 1198701198980 119870119898+1

0isin 1198671(Ω) and 119870

119898(1 minus (119870

119898119870119870))

119870119898+1

(1 minus (119870119898+1

119870119870)) isin 119871

2(0 119879 119871

2(Ω)) we get

119870119898+1

0minus 119870119898

0isin 1198671

(Ω)

[119908119870119870119898+1

(1 minus119870119898+1

119870119870

) minus 119908119870119870119898

(1 minus119870119898

119870119870

)]

isin 1198712

(0 119879 1198712

(Ω))

(87)


055

05

045

04

035

03

025

02

015

01

0050 5 10 15 20 25 30 35 40

119905

119873

120575119873 = 50

120575119873 = 10

120575119873 = 2

(a)

119905 = 10

119873

07

06

05

04

03

02

01

00 01 02 03 04 05 06 07 09 108

120575119873 = 50

120575119873 = 10

120575119873 = 5

120575119873 = 2

120575119873 = 05

(b)



119905 = 1

1090807060504030201

00

051 0 02 04

08061

119910

119870

119909

(a)

119905 = 10

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

(b)

119905 = 50

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(c)

119905 = 1

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(d)

119905 = 10

109

08

0706

06

0504

04

0302

02

0100

1 0 02 040806

1

119910

119870

119909

(e)

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

119905 = 50

(f)



109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(a)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(b)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(c)

109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(d)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(e)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

minus119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

(88)


5= 1198885(Ω 119879) and 119888

6=


int

119879

0

100381710038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

) minus 119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1199082

119870

1198702

119870

100381710038171003817100381710038171003817(119870119898

)2


)2100381710038171003817100381710038171003817

2

1198712(Ω)

+ 1199082

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1198884

5

1199082

119870

1198702

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

10038171003817100381710038171003817119870119898

+ 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

+ 1199082

1198701198882

6

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le (int

119879

0

(41198884

5

1199082

119870

1198702

119870

[10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)

+10038171003817100381710038171003817119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

]

+21199082

1198701198882

6)119889119905)

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le (321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

1198796

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω)

+ 21199082

1198701198882

61198796)

times10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(89)

where

1198796= min 1

81

81205811

8120582

119896 = 321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω) 120582 = 2119908

2

1198701198882

6

(90)


109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(a)

060 02 04

08 106

002

04

04

081

119909

06

002

081

119910

119905 = 10

119873

(b)

109080706

06

0504030201

00

051 0 02 04 08 1

119909119910

119905 = 50

119873

(c)

109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(d)

109080706

06

0504030201

00

005 02 04 08 1119909119910

119905 = 10

119873

(e)

109080706

06

0504030201

100

051 0 02 04 08 1119909

119910

119905 = 50

119873

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le1

4(310038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

(91)


119871infin(0 119879 119871


solution follows


2 1198732 1198672) be two solu-


10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883 (92)

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883(93)

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le1

4

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+1

4

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883

+1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883

(94)


1= 1198672


of the uniqueness








119870 (0 119909) = exp(minus|119909|2

120598) 120598 gt 0


120598)) 120598 gt 0

119867 (0 119909) = 120577119870 (0 119909) 120577 isin [0 1)

(95)




119894 where 119894 = 1 2 119898 + 1 and 119894 = 1 2 (119898 + 1)

2


119867119899+1

119894minus 119867119899

119894

Δ119905= 120575119867119870119899

119894minus 120575119867119867119899+1

119894+ (

119867119899+1

119894minus1minus 2119867119899+1

119894+ 119867119899+1

119894+1

Δ1199092)

(96)


AHH119899+1

= H119899 + 120599119899H (97)




119873119899+1

119894=

1

1 + Δ119905120575119873119867119899+1

119894

[119873119899

119894+ Δ119905119873

119899

119894(1 minus 120579119870

119899

119894minus 119873119899

119894)] (98)



nabla (119863 (119873)nabla119870)|119909119894=

1

2(Δ119909)2sum

119896isin119873119894

(119863 (119873119899+1

119896) + 119863 (119873

119899+1

119894))

times (119870119899+1

119896minus 119870119899+1

119894)

(99)

where119863(119873) = 120575119896(1 minus119873) and119873



119894has two


AKK119899+1

= K119899 + 120599K (100)


119896119870119899

119894(1minus

119870119899

119894minus 119873119899+1


























119867and larger 120575













119873=





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

120572 (119905 x) = int119905

0

119889119873119867119898+1

(V x) 119889V

120573 (119904 x) = 119908119873119873119898

(119904 x) (1 minus 119873119898

(119904 x)119870119873

minus 120579119870119898

(119904 x)119870119870

)

(53)


119873119898+1

isin 119871infin

((0 119879] times Ω) (54)

119873119898+1


((0 119879] times Ω) (55)


Next estimate10038171003817100381710038171003817119873119898+1

119905

10038171003817100381710038171003817119871infin((0119879]timesΩ)

le 119908119873

10038171003817100381710038171198731198981003817100381710038171003817119871infin((0119879]timesΩ)

10038171003817100381710038171003817100381710038171003817

1 minus119873119898

119870119873

minus 120579119870119898

119870119870

10038171003817100381710038171003817100381710038171003817119871infin((0119879]timesΩ)

+ 119889119873

10038171003817100381710038171003817119873119898+110038171003817100381710038171003817119871infin((0119879]timesΩ)

10038171003817100381710038171003817119867119898+110038171003817100381710038171003817119871infin((0119879]timesΩ)

lt infin

(56)



119871infin((0119879]timesΩ)


10038171003817100381710038171003817119873119898+1

119905




119873119898+1

(119905 x) le 119870119873

2119890minus120572(119905x)

+ int

119905

0

120573 (119904 x) 119890minus(120572(119905x)minus120572(119904x))119889119904

le119870119873

2119890minus120572(119905x)

+ 119908119873

119870119873

2int

119905

0

119890minus(120572(119905x)minus120572(119904x))

119889119904

(58)


119867119898+1


estimate

119873119898+1

(119905 x) le 119870119873

2119890minus119889119873119867119905 + 119908

119873

119870119873

2

1

119889119873119867

(1 minus 119890minus119889119873119867119905)

le119870119873

2((1 minus

119908119873

119889119873119867

) 119890minus119889119873119867119905 +

119908119873

119889119873119867

) le119870119873

2

(59)


Due to (52) we get

10038171003817100381710038171003817119873119898+1

(119905)10038171003817100381710038171003817

2

1198671(Ω)

=

10038171003817100381710038171003817100381710038171003817

119890minus120572(119905)

1198730+ 119890minus120572(119905)

int

119905

0

120573(119904)119890120572(119904)

119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 21199082

119873

10038171003817100381710038171003817100381710038171003817

int

119905

0

119873119898

(119904) (1 minus119873119898(119904)

119870119873

minus 120579119870119898(119904)

119870119870

)119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 21199082

119873

100381710038171003817100381710038171003817100381710038171003817

int

119905

0

(119873119898

(119904) minus(119873119898(119904))2

1198702

119873

)119889119904

100381710038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

le 210038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

+ 41199082

119873(

10038171003817100381710038171003817100381710038171003817

int

119905

0

119873119898

(119904)119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

+1

1198702

119873

10038171003817100381710038171003817100381710038171003817

int

119905

0

(119873119898

(119904))2

119889119904

10038171003817100381710038171003817100381710038171003817

2

1198671(Ω)

)

le10038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)[2 + 4119908

2

119873119862 (Ω 119879) 119879

2

]

le C (Ω 119879)10038171003817100381710038171198730

1003817100381710038171003817

2

1198671(Ω)

(60)




int

119879

0

10038171003817100381710038171003817100381710038171003817

119870119898

(1 minus119870119898

119870119870

)

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le int

119879

0

(100381710038171003817100381711987011989810038171003817100381710038171198712(Ω)

+

1003817100381710038171003817100381710038171003817100381710038171003817

(119870119898)2

119870119870

10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

)

2

119889119905

le 2int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198712(Ω)119889119905 + 2int

119879

0

1003817100381710038171003817100381710038171003817100381710038171003817

(119870119898)2

119870119870

1003817100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 21198882

1int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)119889119905 + 2

1198884

2

1198702

119870

int

119879

0

10038171003817100381710038171198701198981003817100381710038171003817

4

1198671(Ω)119889119905

le 81198882

11198622

(Ω 119879)10038171003817100381710038171198700

1003817100381710038171003817

2

1198671(Ω)11987911198792

+ 321198884

2

1198702

119870

1198624

(Ω 119879)10038171003817100381710038171198700

1003817100381710038171003817

4

1198671(Ω)11987911198792lt infin

(61)


2(0 119879 119871

2(Ω)) and119870




119870119898+1

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

119870119898+1

119905isin 1198712

(0 119879 1198712

(Ω))

(62)


11198622(Ω 119879) 119879

11198624(Ω 119879) le 1

and

1198792= min1

2

1

161199082

1198701198882

1

100381710038171003817100381711987001003817100381710038171003817

1198702

119870

641199082

1198701198884

2

100381710038171003817100381711987001003817100381710038171003817

3 (63)

Then

int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905 le10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)(64)


+10038171003817100381710038171003817119870119898+1100381710038171003817100381710038171198712(01198791198672(Ω))

le 2119862 (Ω 119879)10038171003817100381710038171198700

10038171003817100381710038171198671(Ω) (65)


auxiliary function

120585119898+1


(119905 x) (66)


0

119870119898+1

le 119870119870 (67)





0


120597120585119898+1

120597119905minus nabla sdot (119863

119870(1 minus

119873119898+1

119870119873

)nabla120585119898+1

)


(1 minus119870119898

119870119870

)

(68)

Since

119870119898

(1 minus119870119898

119870119870

) le 119870119870 (69)



(1 minus119870119898

119870119870

) ge 0 (70)


3and 120578 such




(119905 x) = 120585119898+1 ge 0 (71)


119870119898+1

le 119870119870 (72)





0and





2(Ω) this



0 119867119898+1

0isin 1198671(Ω)


119867119898+1

0minus 119867119898

0isin 1198671

(Ω)

119870119898+1

minus 119870119898

isin 1198712

(0 119879 1198712

(Ω))

(73)



10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817119908119867119870119898

minus 119908119867119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

(74)


3= 1198883(Ω 119879) it follows that

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879)1199082

1198671198882

3int

119879

0

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le 119862 (Ω 119879)1199082

1198671198882

31198794

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

2

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(75)

where

1198794= min1

4

1

4119862 (Ω 119879)1199082

1198671198882

3

(76)




120597

120597119905(119873119898+1

minus 119873119898

) + 119889119873(119867119898+1

119873119898+1

minus 119867119898

119873119898

)

= 119908119873(119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

))

(77)


119898119870119873)minus(119870

119898119870119870))minus

119873119898minus1

(1 minus (119873119898minus1

119870119873) minus (119870

119898minus1119870119870)))




1

2intΩ

120597

120597119905(119873119898+1

minus 119873119898

)2

119889x

+ 119889119873intΩ

(119873119898+1

minus 119873119898

)2

119867119898+1

119889x

= intΩ

(ℎ (119873119898

119873119898minus1

) minus 119889119873119873119898

(119867119898+1

minus 119867119898

))

times (119873119898+1

minus 119873119898

) 119889x

(78)

Thus

119889

119889119905

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 2119908119873intΩ

10038161003816100381610038161003816100381610038161003816

(119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

))

times (119873119898+1

minus 119873119898

)

10038161003816100381610038161003816100381610038161003816

119889x

+ 2119889119873intΩ

10038161003816100381610038161003816119873119898

(119867119898+1

minus 119867119898

) (119873119898+1

minus 119873119898

)10038161003816100381610038161003816119889x

le [2119908119873

10038171003817100381710038171003817100381710038171003817

119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

)

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

+2119889119873

10038171003817100381710038171003817119873119898

(119867119898+1

minus 119867119898

)100381710038171003817100381710038171198712(Ω)

]

times10038171003817100381710038171003817119873119898+1

minus 119873119898100381710038171003817100381710038171198712(Ω)

(79)




5 times 107

cm3

119870119870

5 times 107

cm3

119908119873

1 times 10minus6

s119908119870

1 times 10minus6

s119863119870

2 times 10minus10

cm2s119863119867

5 times 10minus6

cm2s119908119867


119889119867

11 times 10minus4

s119889119873

0 rarr 10M sdot s

Let (recall (19))

119872max = max 119872119873119898 =

10038171003817100381710038171198731198981003817100381710038171003817119871infin((0119879]timesΩ)

119873119873119898minus1 =


(80)



2119908119873

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119873119898


minus(119873119898)2

119870119873

+

(119873119898minus1

)2

119870119873

minus119873119898119870119898

119870119870

+119873119898minus1

119870119898minus1

119870119870

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

le 2119908119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+4119908119873119872max119870119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 2119908119873119870119873

119870119870

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

+ 2119908119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

le 119862

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 119862

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

(81)

with 119862= 4119908

119873(1 + (119872max119870119873)) and 119862 = 2119908119873119870119873119870119870


2119889119873

10038171003817100381710038171003817119873119898

(119867119898+1

minus 119867119898

)100381710038171003817100381710038171198712(Ω)

le 1198891198731198884

119870119873

2

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

= 119862

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

(82)


119905 = 1

01

01

1

09

08

07

06

05

04

03

02

01090807060503 04020

119909

(a)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 10

119909

(b)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)



119889

119889119905

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le1

2(119862

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 119862

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

+119862

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

)

2

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

(83)


10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198901199052

int

119905

0

(1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

) 119889119904

(84)


01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 1

119909

(a)

01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 10

119909

(b)

01

09

08

07

06

05

04

03

02

001 1

1

090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)



1198622

we get

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le 119863 (Ω 119879) (10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

)1198795

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+5

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

)

(85)


119863 (Ω 119879) 1198795le1

4 (86)

Now since 1198701198980 119870119898+1

0isin 1198671(Ω) and 119870

119898(1 minus (119870

119898119870119870))

119870119898+1

(1 minus (119870119898+1

119870119870)) isin 119871

2(0 119879 119871

2(Ω)) we get

119870119898+1

0minus 119870119898

0isin 1198671

(Ω)

[119908119870119870119898+1

(1 minus119870119898+1

119870119870

) minus 119908119870119870119898

(1 minus119870119898

119870119870

)]

isin 1198712

(0 119879 1198712

(Ω))

(87)


055

05

045

04

035

03

025

02

015

01

0050 5 10 15 20 25 30 35 40

119905

119873

120575119873 = 50

120575119873 = 10

120575119873 = 2

(a)

119905 = 10

119873

07

06

05

04

03

02

01

00 01 02 03 04 05 06 07 09 108

120575119873 = 50

120575119873 = 10

120575119873 = 5

120575119873 = 2

120575119873 = 05

(b)



119905 = 1

1090807060504030201

00

051 0 02 04

08061

119910

119870

119909

(a)

119905 = 10

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

(b)

119905 = 50

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(c)

119905 = 1

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(d)

119905 = 10

109

08

0706

06

0504

04

0302

02

0100

1 0 02 040806

1

119910

119870

119909

(e)

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

119905 = 50

(f)



109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(a)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(b)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(c)

109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(d)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(e)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

minus119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

(88)


5= 1198885(Ω 119879) and 119888

6=


int

119879

0

100381710038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

) minus 119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1199082

119870

1198702

119870

100381710038171003817100381710038171003817(119870119898

)2


)2100381710038171003817100381710038171003817

2

1198712(Ω)

+ 1199082

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1198884

5

1199082

119870

1198702

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

10038171003817100381710038171003817119870119898

+ 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

+ 1199082

1198701198882

6

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le (int

119879

0

(41198884

5

1199082

119870

1198702

119870

[10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)

+10038171003817100381710038171003817119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

]

+21199082

1198701198882

6)119889119905)

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le (321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

1198796

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω)

+ 21199082

1198701198882

61198796)

times10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(89)

where

1198796= min 1

81

81205811

8120582

119896 = 321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω) 120582 = 2119908

2

1198701198882

6

(90)


109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(a)

060 02 04

08 106

002

04

04

081

119909

06

002

081

119910

119905 = 10

119873

(b)

109080706

06

0504030201

00

051 0 02 04 08 1

119909119910

119905 = 50

119873

(c)

109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(d)

109080706

06

0504030201

00

005 02 04 08 1119909119910

119905 = 10

119873

(e)

109080706

06

0504030201

100

051 0 02 04 08 1119909

119910

119905 = 50

119873

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le1

4(310038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

(91)


119871infin(0 119879 119871


solution follows


2 1198732 1198672) be two solu-


10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883 (92)

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883(93)

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le1

4

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+1

4

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883

+1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883

(94)


1= 1198672


of the uniqueness








119870 (0 119909) = exp(minus|119909|2

120598) 120598 gt 0


120598)) 120598 gt 0

119867 (0 119909) = 120577119870 (0 119909) 120577 isin [0 1)

(95)




119894 where 119894 = 1 2 119898 + 1 and 119894 = 1 2 (119898 + 1)

2


119867119899+1

119894minus 119867119899

119894

Δ119905= 120575119867119870119899

119894minus 120575119867119867119899+1

119894+ (

119867119899+1

119894minus1minus 2119867119899+1

119894+ 119867119899+1

119894+1

Δ1199092)

(96)


AHH119899+1

= H119899 + 120599119899H (97)




119873119899+1

119894=

1

1 + Δ119905120575119873119867119899+1

119894

[119873119899

119894+ Δ119905119873

119899

119894(1 minus 120579119870

119899

119894minus 119873119899

119894)] (98)



nabla (119863 (119873)nabla119870)|119909119894=

1

2(Δ119909)2sum

119896isin119873119894

(119863 (119873119899+1

119896) + 119863 (119873

119899+1

119894))

times (119870119899+1

119896minus 119870119899+1

119894)

(99)

where119863(119873) = 120575119896(1 minus119873) and119873



119894has two


AKK119899+1

= K119899 + 120599K (100)


119896119870119899

119894(1minus

119870119899

119894minus 119873119899+1


























119867and larger 120575













119873=





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119870119898+1

isin 1198712

(0 1198791198672


(0 1198791198671

(Ω))

119870119898+1

119905isin 1198712

(0 119879 1198712

(Ω))

(62)


11198622(Ω 119879) 119879

11198624(Ω 119879) le 1

and

1198792= min1

2

1

161199082

1198701198882

1

100381710038171003817100381711987001003817100381710038171003817

1198702

119870

641199082

1198701198884

2

100381710038171003817100381711987001003817100381710038171003817

3 (63)

Then

int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905 le10038171003817100381710038171198700

10038171003817100381710038171198671(Ω)(64)


+10038171003817100381710038171003817119870119898+1100381710038171003817100381710038171198712(01198791198672(Ω))

le 2119862 (Ω 119879)10038171003817100381710038171198700

10038171003817100381710038171198671(Ω) (65)


auxiliary function

120585119898+1


(119905 x) (66)


0

119870119898+1

le 119870119870 (67)





0


120597120585119898+1

120597119905minus nabla sdot (119863

119870(1 minus

119873119898+1

119870119873

)nabla120585119898+1

)


(1 minus119870119898

119870119870

)

(68)

Since

119870119898

(1 minus119870119898

119870119870

) le 119870119870 (69)



(1 minus119870119898

119870119870

) ge 0 (70)


3and 120578 such




(119905 x) = 120585119898+1 ge 0 (71)


119870119898+1

le 119870119870 (72)





0and





2(Ω) this



0 119867119898+1

0isin 1198671(Ω)


119867119898+1

0minus 119867119898

0isin 1198671

(Ω)

119870119898+1

minus 119870119898

isin 1198712

(0 119879 1198712

(Ω))

(73)



10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817119908119867119870119898

minus 119908119867119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

(74)


3= 1198883(Ω 119879) it follows that

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879)1199082

1198671198882

3int

119879

0

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le 119862 (Ω 119879)1199082

1198671198882

31198794

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

2

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(75)

where

1198794= min1

4

1

4119862 (Ω 119879)1199082

1198671198882

3

(76)




120597

120597119905(119873119898+1

minus 119873119898

) + 119889119873(119867119898+1

119873119898+1

minus 119867119898

119873119898

)

= 119908119873(119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

))

(77)


119898119870119873)minus(119870

119898119870119870))minus

119873119898minus1

(1 minus (119873119898minus1

119870119873) minus (119870

119898minus1119870119870)))




1

2intΩ

120597

120597119905(119873119898+1

minus 119873119898

)2

119889x

+ 119889119873intΩ

(119873119898+1

minus 119873119898

)2

119867119898+1

119889x

= intΩ

(ℎ (119873119898

119873119898minus1

) minus 119889119873119873119898

(119867119898+1

minus 119867119898

))

times (119873119898+1

minus 119873119898

) 119889x

(78)

Thus

119889

119889119905

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 2119908119873intΩ

10038161003816100381610038161003816100381610038161003816

(119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

))

times (119873119898+1

minus 119873119898

)

10038161003816100381610038161003816100381610038161003816

119889x

+ 2119889119873intΩ

10038161003816100381610038161003816119873119898

(119867119898+1

minus 119867119898

) (119873119898+1

minus 119873119898

)10038161003816100381610038161003816119889x

le [2119908119873

10038171003817100381710038171003817100381710038171003817

119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

)

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

+2119889119873

10038171003817100381710038171003817119873119898

(119867119898+1

minus 119867119898

)100381710038171003817100381710038171198712(Ω)

]

times10038171003817100381710038171003817119873119898+1

minus 119873119898100381710038171003817100381710038171198712(Ω)

(79)




5 times 107

cm3

119870119870

5 times 107

cm3

119908119873

1 times 10minus6

s119908119870

1 times 10minus6

s119863119870

2 times 10minus10

cm2s119863119867

5 times 10minus6

cm2s119908119867


119889119867

11 times 10minus4

s119889119873

0 rarr 10M sdot s

Let (recall (19))

119872max = max 119872119873119898 =

10038171003817100381710038171198731198981003817100381710038171003817119871infin((0119879]timesΩ)

119873119873119898minus1 =


(80)



2119908119873

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119873119898


minus(119873119898)2

119870119873

+

(119873119898minus1

)2

119870119873

minus119873119898119870119898

119870119870

+119873119898minus1

119870119898minus1

119870119870

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

le 2119908119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+4119908119873119872max119870119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 2119908119873119870119873

119870119870

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

+ 2119908119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

le 119862

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 119862

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

(81)

with 119862= 4119908

119873(1 + (119872max119870119873)) and 119862 = 2119908119873119870119873119870119870


2119889119873

10038171003817100381710038171003817119873119898

(119867119898+1

minus 119867119898

)100381710038171003817100381710038171198712(Ω)

le 1198891198731198884

119870119873

2

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

= 119862

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

(82)


119905 = 1

01

01

1

09

08

07

06

05

04

03

02

01090807060503 04020

119909

(a)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 10

119909

(b)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)



119889

119889119905

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le1

2(119862

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 119862

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

+119862

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

)

2

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

(83)


10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198901199052

int

119905

0

(1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

) 119889119904

(84)


01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 1

119909

(a)

01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 10

119909

(b)

01

09

08

07

06

05

04

03

02

001 1

1

090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)



1198622

we get

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le 119863 (Ω 119879) (10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

)1198795

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+5

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

)

(85)


119863 (Ω 119879) 1198795le1

4 (86)

Now since 1198701198980 119870119898+1

0isin 1198671(Ω) and 119870

119898(1 minus (119870

119898119870119870))

119870119898+1

(1 minus (119870119898+1

119870119870)) isin 119871

2(0 119879 119871

2(Ω)) we get

119870119898+1

0minus 119870119898

0isin 1198671

(Ω)

[119908119870119870119898+1

(1 minus119870119898+1

119870119870

) minus 119908119870119870119898

(1 minus119870119898

119870119870

)]

isin 1198712

(0 119879 1198712

(Ω))

(87)


055

05

045

04

035

03

025

02

015

01

0050 5 10 15 20 25 30 35 40

119905

119873

120575119873 = 50

120575119873 = 10

120575119873 = 2

(a)

119905 = 10

119873

07

06

05

04

03

02

01

00 01 02 03 04 05 06 07 09 108

120575119873 = 50

120575119873 = 10

120575119873 = 5

120575119873 = 2

120575119873 = 05

(b)



119905 = 1

1090807060504030201

00

051 0 02 04

08061

119910

119870

119909

(a)

119905 = 10

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

(b)

119905 = 50

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(c)

119905 = 1

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(d)

119905 = 10

109

08

0706

06

0504

04

0302

02

0100

1 0 02 040806

1

119910

119870

119909

(e)

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

119905 = 50

(f)



109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(a)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(b)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(c)

109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(d)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(e)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

minus119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

(88)


5= 1198885(Ω 119879) and 119888

6=


int

119879

0

100381710038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

) minus 119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1199082

119870

1198702

119870

100381710038171003817100381710038171003817(119870119898

)2


)2100381710038171003817100381710038171003817

2

1198712(Ω)

+ 1199082

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1198884

5

1199082

119870

1198702

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

10038171003817100381710038171003817119870119898

+ 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

+ 1199082

1198701198882

6

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le (int

119879

0

(41198884

5

1199082

119870

1198702

119870

[10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)

+10038171003817100381710038171003817119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

]

+21199082

1198701198882

6)119889119905)

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le (321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

1198796

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω)

+ 21199082

1198701198882

61198796)

times10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(89)

where

1198796= min 1

81

81205811

8120582

119896 = 321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω) 120582 = 2119908

2

1198701198882

6

(90)


109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(a)

060 02 04

08 106

002

04

04

081

119909

06

002

081

119910

119905 = 10

119873

(b)

109080706

06

0504030201

00

051 0 02 04 08 1

119909119910

119905 = 50

119873

(c)

109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(d)

109080706

06

0504030201

00

005 02 04 08 1119909119910

119905 = 10

119873

(e)

109080706

06

0504030201

100

051 0 02 04 08 1119909

119910

119905 = 50

119873

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le1

4(310038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

(91)


119871infin(0 119879 119871


solution follows


2 1198732 1198672) be two solu-


10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883 (92)

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883(93)

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le1

4

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+1

4

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883

+1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883

(94)


1= 1198672


of the uniqueness








119870 (0 119909) = exp(minus|119909|2

120598) 120598 gt 0


120598)) 120598 gt 0

119867 (0 119909) = 120577119870 (0 119909) 120577 isin [0 1)

(95)




119894 where 119894 = 1 2 119898 + 1 and 119894 = 1 2 (119898 + 1)

2


119867119899+1

119894minus 119867119899

119894

Δ119905= 120575119867119870119899

119894minus 120575119867119867119899+1

119894+ (

119867119899+1

119894minus1minus 2119867119899+1

119894+ 119867119899+1

119894+1

Δ1199092)

(96)


AHH119899+1

= H119899 + 120599119899H (97)




119873119899+1

119894=

1

1 + Δ119905120575119873119867119899+1

119894

[119873119899

119894+ Δ119905119873

119899

119894(1 minus 120579119870

119899

119894minus 119873119899

119894)] (98)



nabla (119863 (119873)nabla119870)|119909119894=

1

2(Δ119909)2sum

119896isin119873119894

(119863 (119873119899+1

119896) + 119863 (119873

119899+1

119894))

times (119870119899+1

119896minus 119870119899+1

119894)

(99)

where119863(119873) = 120575119896(1 minus119873) and119873



119894has two


AKK119899+1

= K119899 + 120599K (100)


119896119870119899

119894(1minus

119870119899

119894minus 119873119899+1


























119867and larger 120575













119873=





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120597

120597119905(119873119898+1

minus 119873119898

) + 119889119873(119867119898+1

119873119898+1

minus 119867119898

119873119898

)

= 119908119873(119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

))

(77)


119898119870119873)minus(119870

119898119870119870))minus

119873119898minus1

(1 minus (119873119898minus1

119870119873) minus (119870

119898minus1119870119870)))




1

2intΩ

120597

120597119905(119873119898+1

minus 119873119898

)2

119889x

+ 119889119873intΩ

(119873119898+1

minus 119873119898

)2

119867119898+1

119889x

= intΩ

(ℎ (119873119898

119873119898minus1

) minus 119889119873119873119898

(119867119898+1

minus 119867119898

))

times (119873119898+1

minus 119873119898

) 119889x

(78)

Thus

119889

119889119905

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 2119908119873intΩ

10038161003816100381610038161003816100381610038161003816

(119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

))

times (119873119898+1

minus 119873119898

)

10038161003816100381610038161003816100381610038161003816

119889x

+ 2119889119873intΩ

10038161003816100381610038161003816119873119898

(119867119898+1

minus 119867119898

) (119873119898+1

minus 119873119898

)10038161003816100381610038161003816119889x

le [2119908119873

10038171003817100381710038171003817100381710038171003817

119873119898

(1 minus119873119898

119870119873

minus119870119898

119870119870

)


(1 minus119873119898minus1

119870119873


119870119870

)

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

+2119889119873

10038171003817100381710038171003817119873119898

(119867119898+1

minus 119867119898

)100381710038171003817100381710038171198712(Ω)

]

times10038171003817100381710038171003817119873119898+1

minus 119873119898100381710038171003817100381710038171198712(Ω)

(79)




5 times 107

cm3

119870119870

5 times 107

cm3

119908119873

1 times 10minus6

s119908119870

1 times 10minus6

s119863119870

2 times 10minus10

cm2s119863119867

5 times 10minus6

cm2s119908119867


119889119867

11 times 10minus4

s119889119873

0 rarr 10M sdot s

Let (recall (19))

119872max = max 119872119873119898 =

10038171003817100381710038171198731198981003817100381710038171003817119871infin((0119879]timesΩ)

119873119873119898minus1 =


(80)



2119908119873

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119873119898


minus(119873119898)2

119870119873

+

(119873119898minus1

)2

119870119873

minus119873119898119870119898

119870119870

+119873119898minus1

119870119898minus1

119870119870

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

le 2119908119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+4119908119873119872max119870119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 2119908119873119870119873

119870119870

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

+ 2119908119873

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

le 119862

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 119862

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

(81)

with 119862= 4119908

119873(1 + (119872max119870119873)) and 119862 = 2119908119873119870119873119870119870


2119889119873

10038171003817100381710038171003817119873119898

(119867119898+1

minus 119867119898

)100381710038171003817100381710038171198712(Ω)

le 1198891198731198884

119870119873

2

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

= 119862

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

(82)


119905 = 1

01

01

1

09

08

07

06

05

04

03

02

01090807060503 04020

119909

(a)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 10

119909

(b)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)



119889

119889119905

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le1

2(119862

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 119862

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

+119862

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

)

2

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

(83)


10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198901199052

int

119905

0

(1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

) 119889119904

(84)


01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 1

119909

(a)

01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 10

119909

(b)

01

09

08

07

06

05

04

03

02

001 1

1

090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)



1198622

we get

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le 119863 (Ω 119879) (10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

)1198795

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+5

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

)

(85)


119863 (Ω 119879) 1198795le1

4 (86)

Now since 1198701198980 119870119898+1

0isin 1198671(Ω) and 119870

119898(1 minus (119870

119898119870119870))

119870119898+1

(1 minus (119870119898+1

119870119870)) isin 119871

2(0 119879 119871

2(Ω)) we get

119870119898+1

0minus 119870119898

0isin 1198671

(Ω)

[119908119870119870119898+1

(1 minus119870119898+1

119870119870

) minus 119908119870119870119898

(1 minus119870119898

119870119870

)]

isin 1198712

(0 119879 1198712

(Ω))

(87)


055

05

045

04

035

03

025

02

015

01

0050 5 10 15 20 25 30 35 40

119905

119873

120575119873 = 50

120575119873 = 10

120575119873 = 2

(a)

119905 = 10

119873

07

06

05

04

03

02

01

00 01 02 03 04 05 06 07 09 108

120575119873 = 50

120575119873 = 10

120575119873 = 5

120575119873 = 2

120575119873 = 05

(b)



119905 = 1

1090807060504030201

00

051 0 02 04

08061

119910

119870

119909

(a)

119905 = 10

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

(b)

119905 = 50

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(c)

119905 = 1

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(d)

119905 = 10

109

08

0706

06

0504

04

0302

02

0100

1 0 02 040806

1

119910

119870

119909

(e)

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

119905 = 50

(f)



109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(a)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(b)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(c)

109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(d)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(e)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

minus119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

(88)


5= 1198885(Ω 119879) and 119888

6=


int

119879

0

100381710038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

) minus 119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1199082

119870

1198702

119870

100381710038171003817100381710038171003817(119870119898

)2


)2100381710038171003817100381710038171003817

2

1198712(Ω)

+ 1199082

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1198884

5

1199082

119870

1198702

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

10038171003817100381710038171003817119870119898

+ 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

+ 1199082

1198701198882

6

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le (int

119879

0

(41198884

5

1199082

119870

1198702

119870

[10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)

+10038171003817100381710038171003817119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

]

+21199082

1198701198882

6)119889119905)

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le (321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

1198796

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω)

+ 21199082

1198701198882

61198796)

times10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(89)

where

1198796= min 1

81

81205811

8120582

119896 = 321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω) 120582 = 2119908

2

1198701198882

6

(90)


109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(a)

060 02 04

08 106

002

04

04

081

119909

06

002

081

119910

119905 = 10

119873

(b)

109080706

06

0504030201

00

051 0 02 04 08 1

119909119910

119905 = 50

119873

(c)

109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(d)

109080706

06

0504030201

00

005 02 04 08 1119909119910

119905 = 10

119873

(e)

109080706

06

0504030201

100

051 0 02 04 08 1119909

119910

119905 = 50

119873

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le1

4(310038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

(91)


119871infin(0 119879 119871


solution follows


2 1198732 1198672) be two solu-


10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883 (92)

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883(93)

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le1

4

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+1

4

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883

+1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883

(94)


1= 1198672


of the uniqueness








119870 (0 119909) = exp(minus|119909|2

120598) 120598 gt 0


120598)) 120598 gt 0

119867 (0 119909) = 120577119870 (0 119909) 120577 isin [0 1)

(95)




119894 where 119894 = 1 2 119898 + 1 and 119894 = 1 2 (119898 + 1)

2


119867119899+1

119894minus 119867119899

119894

Δ119905= 120575119867119870119899

119894minus 120575119867119867119899+1

119894+ (

119867119899+1

119894minus1minus 2119867119899+1

119894+ 119867119899+1

119894+1

Δ1199092)

(96)


AHH119899+1

= H119899 + 120599119899H (97)




119873119899+1

119894=

1

1 + Δ119905120575119873119867119899+1

119894

[119873119899

119894+ Δ119905119873

119899

119894(1 minus 120579119870

119899

119894minus 119873119899

119894)] (98)



nabla (119863 (119873)nabla119870)|119909119894=

1

2(Δ119909)2sum

119896isin119873119894

(119863 (119873119899+1

119896) + 119863 (119873

119899+1

119894))

times (119870119899+1

119896minus 119870119899+1

119894)

(99)

where119863(119873) = 120575119896(1 minus119873) and119873



119894has two


AKK119899+1

= K119899 + 120599K (100)


119896119870119899

119894(1minus

119870119899

119894minus 119873119899+1


























119867and larger 120575













119873=





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119905 = 1

01

01

1

09

08

07

06

05

04

03

02

01090807060503 04020

119909

(a)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 10

119909

(b)

01

1

09

08

07

06

05

04

03

02

001 1090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)



119889

119889119905

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le1

2(119862

10038171003817100381710038171003817119873119898

minus 119873119898minus1100381710038171003817100381710038171198712(Ω)

+ 119862

10038171003817100381710038171003817119870119898

minus 119870119898minus1100381710038171003817100381710038171198712(Ω)

+119862

10038171003817100381710038171003817119867119898+1

minus 119867119898100381710038171003817100381710038171198671(Ω)

)

2

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

+1

2

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

(83)


10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

1198712(Ω)

le 1198901199052

int

119905

0

(1198622

10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

1198712(Ω)

+ 1198622

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

+1198622

10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

1198671(Ω)

) 119889119904

(84)


01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 1

119909

(a)

01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 10

119909

(b)

01

09

08

07

06

05

04

03

02

001 1

1

090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)



1198622

we get

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le 119863 (Ω 119879) (10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

)1198795

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+5

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

)

(85)


119863 (Ω 119879) 1198795le1

4 (86)

Now since 1198701198980 119870119898+1

0isin 1198671(Ω) and 119870

119898(1 minus (119870

119898119870119870))

119870119898+1

(1 minus (119870119898+1

119870119870)) isin 119871

2(0 119879 119871

2(Ω)) we get

119870119898+1

0minus 119870119898

0isin 1198671

(Ω)

[119908119870119870119898+1

(1 minus119870119898+1

119870119870

) minus 119908119870119870119898

(1 minus119870119898

119870119870

)]

isin 1198712

(0 119879 1198712

(Ω))

(87)


055

05

045

04

035

03

025

02

015

01

0050 5 10 15 20 25 30 35 40

119905

119873

120575119873 = 50

120575119873 = 10

120575119873 = 2

(a)

119905 = 10

119873

07

06

05

04

03

02

01

00 01 02 03 04 05 06 07 09 108

120575119873 = 50

120575119873 = 10

120575119873 = 5

120575119873 = 2

120575119873 = 05

(b)



119905 = 1

1090807060504030201

00

051 0 02 04

08061

119910

119870

119909

(a)

119905 = 10

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

(b)

119905 = 50

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(c)

119905 = 1

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(d)

119905 = 10

109

08

0706

06

0504

04

0302

02

0100

1 0 02 040806

1

119910

119870

119909

(e)

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

119905 = 50

(f)



109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(a)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(b)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(c)

109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(d)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(e)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

minus119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

(88)


5= 1198885(Ω 119879) and 119888

6=


int

119879

0

100381710038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

) minus 119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1199082

119870

1198702

119870

100381710038171003817100381710038171003817(119870119898

)2


)2100381710038171003817100381710038171003817

2

1198712(Ω)

+ 1199082

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1198884

5

1199082

119870

1198702

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

10038171003817100381710038171003817119870119898

+ 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

+ 1199082

1198701198882

6

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le (int

119879

0

(41198884

5

1199082

119870

1198702

119870

[10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)

+10038171003817100381710038171003817119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

]

+21199082

1198701198882

6)119889119905)

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le (321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

1198796

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω)

+ 21199082

1198701198882

61198796)

times10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(89)

where

1198796= min 1

81

81205811

8120582

119896 = 321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω) 120582 = 2119908

2

1198701198882

6

(90)


109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(a)

060 02 04

08 106

002

04

04

081

119909

06

002

081

119910

119905 = 10

119873

(b)

109080706

06

0504030201

00

051 0 02 04 08 1

119909119910

119905 = 50

119873

(c)

109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(d)

109080706

06

0504030201

00

005 02 04 08 1119909119910

119905 = 10

119873

(e)

109080706

06

0504030201

100

051 0 02 04 08 1119909

119910

119905 = 50

119873

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le1

4(310038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

(91)


119871infin(0 119879 119871


solution follows


2 1198732 1198672) be two solu-


10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883 (92)

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883(93)

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le1

4

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+1

4

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883

+1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883

(94)


1= 1198672


of the uniqueness








119870 (0 119909) = exp(minus|119909|2

120598) 120598 gt 0


120598)) 120598 gt 0

119867 (0 119909) = 120577119870 (0 119909) 120577 isin [0 1)

(95)




119894 where 119894 = 1 2 119898 + 1 and 119894 = 1 2 (119898 + 1)

2


119867119899+1

119894minus 119867119899

119894

Δ119905= 120575119867119870119899

119894minus 120575119867119867119899+1

119894+ (

119867119899+1

119894minus1minus 2119867119899+1

119894+ 119867119899+1

119894+1

Δ1199092)

(96)


AHH119899+1

= H119899 + 120599119899H (97)




119873119899+1

119894=

1

1 + Δ119905120575119873119867119899+1

119894

[119873119899

119894+ Δ119905119873

119899

119894(1 minus 120579119870

119899

119894minus 119873119899

119894)] (98)



nabla (119863 (119873)nabla119870)|119909119894=

1

2(Δ119909)2sum

119896isin119873119894

(119863 (119873119899+1

119896) + 119863 (119873

119899+1

119894))

times (119870119899+1

119896minus 119870119899+1

119894)

(99)

where119863(119873) = 120575119896(1 minus119873) and119873



119894has two


AKK119899+1

= K119899 + 120599K (100)


119896119870119899

119894(1minus

119870119899

119894minus 119873119899+1


























119867and larger 120575













119873=





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 1

119909

(a)

01

09

08

07

06

05

04

03

02

0

1

01 1090807060503 04020

119905 = 10

119909

(b)

01

09

08

07

06

05

04

03

02

001 1

1

090807060503 04020

119905 = 30

119909

(c)

01 1090807060503 04020

14

12

1

08

06

04

02

0

119905 = 50

119909

(d)



1198622

we get

10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le 119863 (Ω 119879) (10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

)1198795

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

le1

4(10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+5

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

)

(85)


119863 (Ω 119879) 1198795le1

4 (86)

Now since 1198701198980 119870119898+1

0isin 1198671(Ω) and 119870

119898(1 minus (119870

119898119870119870))

119870119898+1

(1 minus (119870119898+1

119870119870)) isin 119871

2(0 119879 119871

2(Ω)) we get

119870119898+1

0minus 119870119898

0isin 1198671

(Ω)

[119908119870119870119898+1

(1 minus119870119898+1

119870119870

) minus 119908119870119870119898

(1 minus119870119898

119870119870

)]

isin 1198712

(0 119879 1198712

(Ω))

(87)


055

05

045

04

035

03

025

02

015

01

0050 5 10 15 20 25 30 35 40

119905

119873

120575119873 = 50

120575119873 = 10

120575119873 = 2

(a)

119905 = 10

119873

07

06

05

04

03

02

01

00 01 02 03 04 05 06 07 09 108

120575119873 = 50

120575119873 = 10

120575119873 = 5

120575119873 = 2

120575119873 = 05

(b)



119905 = 1

1090807060504030201

00

051 0 02 04

08061

119910

119870

119909

(a)

119905 = 10

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

(b)

119905 = 50

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(c)

119905 = 1

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(d)

119905 = 10

109

08

0706

06

0504

04

0302

02

0100

1 0 02 040806

1

119910

119870

119909

(e)

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

119905 = 50

(f)



109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(a)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(b)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(c)

109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(d)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(e)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

minus119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

(88)


5= 1198885(Ω 119879) and 119888

6=


int

119879

0

100381710038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

) minus 119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1199082

119870

1198702

119870

100381710038171003817100381710038171003817(119870119898

)2


)2100381710038171003817100381710038171003817

2

1198712(Ω)

+ 1199082

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1198884

5

1199082

119870

1198702

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

10038171003817100381710038171003817119870119898

+ 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

+ 1199082

1198701198882

6

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le (int

119879

0

(41198884

5

1199082

119870

1198702

119870

[10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)

+10038171003817100381710038171003817119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

]

+21199082

1198701198882

6)119889119905)

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le (321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

1198796

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω)

+ 21199082

1198701198882

61198796)

times10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(89)

where

1198796= min 1

81

81205811

8120582

119896 = 321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω) 120582 = 2119908

2

1198701198882

6

(90)


109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(a)

060 02 04

08 106

002

04

04

081

119909

06

002

081

119910

119905 = 10

119873

(b)

109080706

06

0504030201

00

051 0 02 04 08 1

119909119910

119905 = 50

119873

(c)

109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(d)

109080706

06

0504030201

00

005 02 04 08 1119909119910

119905 = 10

119873

(e)

109080706

06

0504030201

100

051 0 02 04 08 1119909

119910

119905 = 50

119873

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le1

4(310038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

(91)


119871infin(0 119879 119871


solution follows


2 1198732 1198672) be two solu-


10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883 (92)

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883(93)

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le1

4

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+1

4

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883

+1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883

(94)


1= 1198672


of the uniqueness








119870 (0 119909) = exp(minus|119909|2

120598) 120598 gt 0


120598)) 120598 gt 0

119867 (0 119909) = 120577119870 (0 119909) 120577 isin [0 1)

(95)




119894 where 119894 = 1 2 119898 + 1 and 119894 = 1 2 (119898 + 1)

2


119867119899+1

119894minus 119867119899

119894

Δ119905= 120575119867119870119899

119894minus 120575119867119867119899+1

119894+ (

119867119899+1

119894minus1minus 2119867119899+1

119894+ 119867119899+1

119894+1

Δ1199092)

(96)


AHH119899+1

= H119899 + 120599119899H (97)




119873119899+1

119894=

1

1 + Δ119905120575119873119867119899+1

119894

[119873119899

119894+ Δ119905119873

119899

119894(1 minus 120579119870

119899

119894minus 119873119899

119894)] (98)



nabla (119863 (119873)nabla119870)|119909119894=

1

2(Δ119909)2sum

119896isin119873119894

(119863 (119873119899+1

119896) + 119863 (119873

119899+1

119894))

times (119870119899+1

119896minus 119870119899+1

119894)

(99)

where119863(119873) = 120575119896(1 minus119873) and119873



119894has two


AKK119899+1

= K119899 + 120599K (100)


119896119870119899

119894(1minus

119870119899

119894minus 119873119899+1


























119867and larger 120575













119873=





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










055

05

045

04

035

03

025

02

015

01

0050 5 10 15 20 25 30 35 40

119905

119873

120575119873 = 50

120575119873 = 10

120575119873 = 2

(a)

119905 = 10

119873

07

06

05

04

03

02

01

00 01 02 03 04 05 06 07 09 108

120575119873 = 50

120575119873 = 10

120575119873 = 5

120575119873 = 2

120575119873 = 05

(b)



119905 = 1

1090807060504030201

00

051 0 02 04

08061

119910

119870

119909

(a)

119905 = 10

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

(b)

119905 = 50

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(c)

119905 = 1

1090807060504030201

0005

1 0 02 040806

1

119910

119870

119909

(d)

119905 = 10

109

08

0706

06

0504

04

0302

02

0100

1 0 02 040806

1

119910

119870

119909

(e)

1090807060504030201

00

051 0 02 04 0806 1

119910

119870

119909

119905 = 50

(f)



109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(a)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(b)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(c)

109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(d)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(e)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

minus119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

(88)


5= 1198885(Ω 119879) and 119888

6=


int

119879

0

100381710038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

) minus 119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1199082

119870

1198702

119870

100381710038171003817100381710038171003817(119870119898

)2


)2100381710038171003817100381710038171003817

2

1198712(Ω)

+ 1199082

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1198884

5

1199082

119870

1198702

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

10038171003817100381710038171003817119870119898

+ 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

+ 1199082

1198701198882

6

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le (int

119879

0

(41198884

5

1199082

119870

1198702

119870

[10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)

+10038171003817100381710038171003817119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

]

+21199082

1198701198882

6)119889119905)

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le (321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

1198796

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω)

+ 21199082

1198701198882

61198796)

times10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(89)

where

1198796= min 1

81

81205811

8120582

119896 = 321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω) 120582 = 2119908

2

1198701198882

6

(90)


109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(a)

060 02 04

08 106

002

04

04

081

119909

06

002

081

119910

119905 = 10

119873

(b)

109080706

06

0504030201

00

051 0 02 04 08 1

119909119910

119905 = 50

119873

(c)

109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(d)

109080706

06

0504030201

00

005 02 04 08 1119909119910

119905 = 10

119873

(e)

109080706

06

0504030201

100

051 0 02 04 08 1119909

119910

119905 = 50

119873

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le1

4(310038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

(91)


119871infin(0 119879 119871


solution follows


2 1198732 1198672) be two solu-


10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883 (92)

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883(93)

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le1

4

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+1

4

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883

+1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883

(94)


1= 1198672


of the uniqueness








119870 (0 119909) = exp(minus|119909|2

120598) 120598 gt 0


120598)) 120598 gt 0

119867 (0 119909) = 120577119870 (0 119909) 120577 isin [0 1)

(95)




119894 where 119894 = 1 2 119898 + 1 and 119894 = 1 2 (119898 + 1)

2


119867119899+1

119894minus 119867119899

119894

Δ119905= 120575119867119870119899

119894minus 120575119867119867119899+1

119894+ (

119867119899+1

119894minus1minus 2119867119899+1

119894+ 119867119899+1

119894+1

Δ1199092)

(96)


AHH119899+1

= H119899 + 120599119899H (97)




119873119899+1

119894=

1

1 + Δ119905120575119873119867119899+1

119894

[119873119899

119894+ Δ119905119873

119899

119894(1 minus 120579119870

119899

119894minus 119873119899

119894)] (98)



nabla (119863 (119873)nabla119870)|119909119894=

1

2(Δ119909)2sum

119896isin119873119894

(119863 (119873119899+1

119896) + 119863 (119873

119899+1

119894))

times (119870119899+1

119896minus 119870119899+1

119894)

(99)

where119863(119873) = 120575119896(1 minus119873) and119873



119894has two


AKK119899+1

= K119899 + 120599K (100)


119896119870119899

119894(1minus

119870119899

119894minus 119873119899+1


























119867and larger 120575













119873=





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(a)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(b)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(c)

109080706

06

0504030201

100

051 0 02 04 08 1

119905 = 1

119909 119910

119867

(d)

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 10

119867

(e)

119867

109080706

06

0504030201

100

051 0 02 04 08 1

119909 119910

119905 = 50

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

le 119862 (Ω 119879) int

119879

0

10038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

)

minus119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

(88)


5= 1198885(Ω 119879) and 119888

6=


int

119879

0

100381710038171003817100381710038171003817100381710038171003817

119908119870119870119898

(1 minus119870119898

119870119870

) minus 119908119870119870119898minus1

(1 minus119870119898minus1

119870119870

)

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1199082

119870

1198702

119870

100381710038171003817100381710038171003817(119870119898

)2


)2100381710038171003817100381710038171003817

2

1198712(Ω)

+ 1199082

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198712(Ω)

119889119905

le 2int

119879

0

1198884

5

1199082

119870

1198702

119870

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

10038171003817100381710038171003817119870119898

+ 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

+ 1199082

1198701198882

6

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

119889119905

le (int

119879

0

(41198884

5

1199082

119870

1198702

119870

[10038171003817100381710038171198701198981003817100381710038171003817

2

1198671(Ω)

+10038171003817100381710038171003817119870119898minus110038171003817100381710038171003817

2

1198671(Ω)

]

+21199082

1198701198882

6)119889119905)

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le (321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

1198796

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω)

+ 21199082

1198701198882

61198796)

times10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

le1

4

10038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

(89)

where

1198796= min 1

81

81205811

8120582

119896 = 321198622

(Ω 119879) 1198884

5

1199082

119870

1198702

119870

100381710038171003817100381711987001003817100381710038171003817

2

1198671(Ω) 120582 = 2119908

2

1198701198882

6

(90)


109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(a)

060 02 04

08 106

002

04

04

081

119909

06

002

081

119910

119905 = 10

119873

(b)

109080706

06

0504030201

00

051 0 02 04 08 1

119909119910

119905 = 50

119873

(c)

109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(d)

109080706

06

0504030201

00

005 02 04 08 1119909119910

119905 = 10

119873

(e)

109080706

06

0504030201

100

051 0 02 04 08 1119909

119910

119905 = 50

119873

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le1

4(310038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

(91)


119871infin(0 119879 119871


solution follows


2 1198732 1198672) be two solu-


10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883 (92)

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883(93)

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le1

4

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+1

4

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883

+1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883

(94)


1= 1198672


of the uniqueness








119870 (0 119909) = exp(minus|119909|2

120598) 120598 gt 0


120598)) 120598 gt 0

119867 (0 119909) = 120577119870 (0 119909) 120577 isin [0 1)

(95)




119894 where 119894 = 1 2 119898 + 1 and 119894 = 1 2 (119898 + 1)

2


119867119899+1

119894minus 119867119899

119894

Δ119905= 120575119867119870119899

119894minus 120575119867119867119899+1

119894+ (

119867119899+1

119894minus1minus 2119867119899+1

119894+ 119867119899+1

119894+1

Δ1199092)

(96)


AHH119899+1

= H119899 + 120599119899H (97)




119873119899+1

119894=

1

1 + Δ119905120575119873119867119899+1

119894

[119873119899

119894+ Δ119905119873

119899

119894(1 minus 120579119870

119899

119894minus 119873119899

119894)] (98)



nabla (119863 (119873)nabla119870)|119909119894=

1

2(Δ119909)2sum

119896isin119873119894

(119863 (119873119899+1

119896) + 119863 (119873

119899+1

119894))

times (119870119899+1

119896minus 119870119899+1

119894)

(99)

where119863(119873) = 120575119896(1 minus119873) and119873



119894has two


AKK119899+1

= K119899 + 120599K (100)


119896119870119899

119894(1minus

119870119899

119894minus 119873119899+1


























119867and larger 120575













119873=





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(a)

060 02 04

08 106

002

04

04

081

119909

06

002

081

119910

119905 = 10

119873

(b)

109080706

06

0504030201

00

051 0 02 04 08 1

119909119910

119905 = 50

119873

(c)

109080706

06

0504030201

00

051 0 02 04 08 1119909

119910

119905 = 1

119873

(d)

109080706

06

0504030201

00

005 02 04 08 1119909119910

119905 = 10

119873

(e)

109080706

06

0504030201

100

051 0 02 04 08 1119909

119910

119905 = 50

119873

(f)



10038171003817100381710038171003817119870119898+1

minus 11987011989810038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898+1

minus 11987311989810038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+10038171003817100381710038171003817119867119898+1

minus 11986711989810038171003817100381710038171003817

2

119883

le1

4(310038171003817100381710038171003817119870119898

minus 119870119898minus110038171003817100381710038171003817

2

119883

+10038171003817100381710038171003817119873119898

minus 119873119898minus110038171003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

)

(91)


119871infin(0 119879 119871


solution follows


2 1198732 1198672) be two solu-


10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883 (92)

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883le1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883(93)

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

le1

4

10038171003817100381710038171198731 minus 11987321003817100381710038171003817

2

119871infin(0119879119871

2(Ω))

+1

4

10038171003817100381710038171198671 minus 11986721003817100381710038171003817

2

119883

+1

4

10038171003817100381710038171198701 minus 11987021003817100381710038171003817

2

119883

(94)


1= 1198672


of the uniqueness








119870 (0 119909) = exp(minus|119909|2

120598) 120598 gt 0


120598)) 120598 gt 0

119867 (0 119909) = 120577119870 (0 119909) 120577 isin [0 1)

(95)




119894 where 119894 = 1 2 119898 + 1 and 119894 = 1 2 (119898 + 1)

2


119867119899+1

119894minus 119867119899

119894

Δ119905= 120575119867119870119899

119894minus 120575119867119867119899+1

119894+ (

119867119899+1

119894minus1minus 2119867119899+1

119894+ 119867119899+1

119894+1

Δ1199092)

(96)


AHH119899+1

= H119899 + 120599119899H (97)




119873119899+1

119894=

1

1 + Δ119905120575119873119867119899+1

119894

[119873119899

119894+ Δ119905119873

119899

119894(1 minus 120579119870

119899

119894minus 119873119899

119894)] (98)



nabla (119863 (119873)nabla119870)|119909119894=

1

2(Δ119909)2sum

119896isin119873119894

(119863 (119873119899+1

119896) + 119863 (119873

119899+1

119894))

times (119870119899+1

119896minus 119870119899+1

119894)

(99)

where119863(119873) = 120575119896(1 minus119873) and119873



119894has two


AKK119899+1

= K119899 + 120599K (100)


119896119870119899

119894(1minus

119870119899

119894minus 119873119899+1


























119867and larger 120575













119873=





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119870 (0 119909) = exp(minus|119909|2

120598) 120598 gt 0


120598)) 120598 gt 0

119867 (0 119909) = 120577119870 (0 119909) 120577 isin [0 1)

(95)




119894 where 119894 = 1 2 119898 + 1 and 119894 = 1 2 (119898 + 1)

2


119867119899+1

119894minus 119867119899

119894

Δ119905= 120575119867119870119899

119894minus 120575119867119867119899+1

119894+ (

119867119899+1

119894minus1minus 2119867119899+1

119894+ 119867119899+1

119894+1

Δ1199092)

(96)


AHH119899+1

= H119899 + 120599119899H (97)




119873119899+1

119894=

1

1 + Δ119905120575119873119867119899+1

119894

[119873119899

119894+ Δ119905119873

119899

119894(1 minus 120579119870

119899

119894minus 119873119899

119894)] (98)



nabla (119863 (119873)nabla119870)|119909119894=

1

2(Δ119909)2sum

119896isin119873119894

(119863 (119873119899+1

119896) + 119863 (119873

119899+1

119894))

times (119870119899+1

119896minus 119870119899+1

119894)

(99)

where119863(119873) = 120575119896(1 minus119873) and119873



119894has two


AKK119899+1

= K119899 + 120599K (100)


119896119870119899

119894(1minus

119870119899

119894minus 119873119899+1


























119867and larger 120575













119873=





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119873=





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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