Research ArticleGlobal Solutions in the Species Competitive ChemotaxisSystem with Inequal Diffusion Rates

Huaihuo Cao12

1College of Mathematics and Statistics Northwest Normal University Lanzhou 730070 China2School of Mathematics and Computer Chizhou College Chizhou 247000 China

Correspondence should be addressed to Huaihuo Cao caguhhaliyuncom

Received 10 July 2016 Revised 16 October 2016 Accepted 23 October 2016

Academic Editor Rigoberto Medina

Copyright copy 2016 Huaihuo CaoThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is devoted to studying the two-species competitive chemotaxis system with signal-dependent chemotactic sensitivitiesand inequal diffusion rates 1199061119905 = Δ1199061 minus nabla sdot (11990611205941(V)nablaV) + 12058311199061(1 minus 1199061 minus 11988611199062) 119909 isin Ω 119905 gt 0 1199062119905 = Δ1199062 minus nabla sdot (11990621205942(V)nablaV) +12058321199062(1 minus 11988621199061 minus 1199062) 119909 isin Ω 119905 gt 0 V119905 = 120591ΔV minus 120574V + 1199061 + 1199062 119909 isin Ω 119905 gt 0 under homogeneous Neumann boundary conditions ina bounded and regular domain Ω sub R119899 (119899 ge 1) If the nonnegative initial date (11990610 11990620 V0) isin (1198621(Ω))3 and V0 isin (V V) where theconstants V gt V ge 0 the system possesses a unique global solution that is uniformly bounded under some suitable assumptions onthe chemotaxis sensitivity functions 1205941(V) 1205942(V) and linear chemical production function minus120574V + 1199061 + 1199062

1 Introduction

In this paper we consider the following two-species compet-itive chemotaxis system with signal-dependent chemotacticsensitivities and inequal diffusion rates

1199061119905 = Δ1199061 minus nabla sdot (11990611205941 (V) nablaV)+ 12058311199061 (1 minus 1199061 minus 11988611199062) 119909 isin Ω 119905 gt 0

1199062119905 = Δ1199062 minus nabla sdot (11990621205942 (V) nablaV)+ 12058321199062 (1 minus 11988621199061 minus 1199062) 119909 isin Ω 119905 gt 0

V119905 = 120591ΔV minus 120574V + 1199061 + 1199062 119909 isin Ω 119905 gt 01205971199061120597] = 1205971199062120597] = 120597V120597] = 0 119909 isin 120597Ω 119905 gt 0

1199061 (119909 0) = 11990610 (119909) 1199062 (119909 0) = 11990620 (119909) V (119909 0) = V0 (119909)

119909 isin Ω

(1)

where Ω is a bounded and regular domain in R119899 (119899 ge 1)and ] is the outward unit normal vector of the boundary 120597Ω1199061(119909 119905) and 1199062(119909 119905) represent the populations densities andboth populations reproduce themselves and mutually com-pete with the other according to the classical Lotka-Volterrakinetics [1] and the diffusion rates of the populations are 1V(119909 119905) denotes the concentration of the chemoattractant andthe diffusion rate of the chemical substance is strictly less than1 (ie 0 lt 120591 lt 1) 1205831 1205832 1198861 1198862 and 120574 are positive parameterswhere 1205831 1205832 are the growth coefficients and 1198861 1198862 are thecompetitive degradation rates of population respectivelyThechemotactic sensitivity function 120594119894(V) isin 1198821infinloc (R+) cap 1198621(R+)for 119894 = 1 2 which is assumed to be positive From abiological point of view when 120594119894(V) gt 0 populations exhibita tendency to move towards higher signal concentrations(chemoattraction) while conversely the choice 120594119894(V) lt 0leads to a model for chemorepulsion where populationsprefer to move away from the chemical in question [2]Denote ℎ(1199061 1199062 V) = minus120574V + 1199061 + 1199062 representing the balancebetween the production of the chemical substance by thepopulations themselves and its natural degradation (see [3]for details)

The classical chemotaxis model was first introduced byKeller and Segel using amathematicalmodel of two parabolic

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 5015246 9 pageshttpdxdoiorg10115520165015246

2 Discrete Dynamics in Nature and Society

equations to describe the aggregation of Dictyostelium dis-coideum as well as a soil-living amoeba in the early 1970s [4]After the pioneeringworks of Keller and Segel a large amountof chemotaxismodels has been used tomodel the phenomenafor population dynamics or gravitational collapse amongothers Winkler [5] studied the chemotaxis system

119906119905 = Δ119906 minus nabla sdot (119906120594 (V) nablaV) 119909 isin Ω 119905 gt 0V119905 = ΔV minus V + 119906 119909 isin Ω 119905 gt 0 (2)

under homogeneous Neumann boundary conditions in asmooth bounded domain Ω sub R119899 It was proved that thechemotactic collapse was absented for any nonnegative initialdate 119906(sdot 0) isin 1198620(Ω) and V(sdot 0) isin 1198821119903(Ω) with some119903 gt 119899 the corresponding initial-boundary value problempossessed a unique global uniformly bounded solution Telloand Winkler [6] considered the parabolic-parabolic-ellipticsystem

119906119905 = Δ119906 minus 1205941nabla sdot (119906nabla119908) + 1205831119906 (1 minus 119906 minus 1198861V) 119909 isin Ω 119905 gt 0

V119905 = ΔV minus 1205942nabla sdot (Vnabla119908) + 1205832V (1 minus 1198862119906 minus V) 119909 isin Ω 119905 gt 0

minusΔ119908 = minus120582119908 + 119906 + V 119909 isin Ω 119905 gt 0

(3)

under homogeneous Neumann boundary conditions in abounded domain Ω sub R119899 (119899 ge 1) with smooth boundaryGiven the suitable positive parameters 1198861 1198862 and 1205941 1205942 theyshowed that all solutions stabilized towards a uniquely deter-mined spatially homogeneous positive steady state within acertain nonempty range of the logistic growth coefficients1205831 and 1205832 Negreanu and Tello [7] studied a two-specieschemotaxis system with nondiffusive chemoattractant

119906119905 = Δ119906 minus nabla sdot (1199061205941 (119908) nabla119908) + 1205831119906 (1 minus 119906) 119909 isin Ω 119905 gt 0

V119905 = ΔV minus nabla sdot (V1205942 (119908) nabla119908) + 1205832V (1 minus V) 119909 isin Ω 119905 gt 0

119908119905 = ℎ (119906 V 119908) 119909 isin Ω 119905 gt 0

(4)

under suitable boundary and initial conditions in an 119899-dimensional open and bounded domain Ω for 119899 ge 1 Theyconsidered the case of positive chemosensitivities and chem-ical production function ℎ increasing as the concentrationof the species 119906 V increasing The paper proved the globalexistence and uniform boundedness of solutions and theasymptotic stability of the spatially homogeneous steady statewas a consequence of the growth of ℎ 120594119894 and the size of 120583119894 for119894 = 1 2

Reviewing the recent studies Zhang andLi [8] consideredthe following fully parabolic system

119906119905 = Δ119906 minus nabla sdot (1199061205941 (119908) nabla119908) + 1205831119906 (1 minus 119906 minus 1198861V) 119909 isin Ω 119905 gt 0

V119905 = ΔV minus nabla sdot (V1205942 (119908) nabla119908) + 1205832V (1 minus 1198862119906 minus V) 119909 isin Ω 119905 gt 0

119908119905 = Δ119908 minus 119908 + 119906 + V 119909 isin Ω 119905 gt 0

(5)

under homogeneous Neumann boundary conditions in asmooth bounded domain Ω sub R119899 (119899 ge 1) By extendingthe method in [5] (see also [9 10]) the first step is to esti-mate some associated weighted functions which depend onsignal density and the second step is to obtain 119871infin-boundsof solutions from 119871119901-bounds using the variation-of-constantsrepresentation and a series of standard semigroup arguments(see [2 5 11 12]) They proved that if the nonnegative initialdate (119906(sdot 0) V(sdot 0)) isin (1198620(Ω))2 and 119908(sdot 0) isin 1198821119903(Ω) forsome 119903 gt 119899 the systempossesses a unique global solution thatis uniformly bounded under some appropriate conditions onthe coefficients 1205831 1205832 and the chemotaxis sensitivity func-tions 1205941(119908) 1205942(119908)

Inspired by the foregoing research the main purposeof this paper is to consider the existence of global solutionfor the two-species competitive chemotaxis model (1) withinequal diffusion rates This paper is organized as followsIn Section 2 we formulate the main results of this paper bymeans of the theorem and establish some preliminarieswhichare important for our proofs In Section 3 we firstly considerthe local existence of solutions and then proceed with theextensibility criterion Finally under some appropriate con-ditions we prove that the solutions are uniformly boundedin time using an iterative method

2 Preliminaries and Main Results

For convenience we denote that 1198921(1199061 1199062) = 1 minus 1199061 minus 119886111990621198922(1199061 1199062) = 1 minus 11988621199061 minus 1199062 and they are quasi-monotonicdecrease in R2+ Denote ℎ(1199061 1199062 V) = minus120574V + 1199061 + 1199062

In order to establish the global existence and uniformboundedness of solutions to (1) we need to make somerestrictive conditions throughout this paper in ℎ(1199061 1199062 V)and 120594119894(V) (119894 = 1 2)

(i) Let the initial date V0 satisfy 0 le V lt V0 lt V where Vand V are some positive constants

(ii) There exist positive constants 1205811 and 1205812 such that

120574V120594119894 (V) le 120581119894 for V le V le V 119894 = 1 2 (6)

(iii) There exist positive constants 12058101 and 12058102 such that

0 lt 1205810119894 le 120594119894 (V) exp(intV

V120594119894 (119904) 119889119904)

for V le V 119894 = 1 2(7)

Discrete Dynamics in Nature and Society 3

(iv) Assume that

ℎ (1199061 1199062 V) ge 0ℎ (1199061 1199062 V) lt 0

for 0 le 1199061 le 1199061 0 le 1199062 le 1199062(8)

where

1199061 fl 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

1199062 fl 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(9)

for 119891119894(V) defined by

119891119894 (V) = exp( 11 minus 120591 intV

V120594119894 (119904) 119889119904) 119894 = 1 2 (10)

(v) Finally for some technical reasons we also assumethat

1205942119894 (V) + (1 minus 120591) 1205941015840119894 (V) le 0 119894 = 1 2 (11)

We illustrate the validity to above assumptions with thefollowing generalized example [7]

Example 1 We take the chemosensitivity functions 120594119894(V) =120572119894(120573119894 + V) (119894 = 1 2) for positive constants 120572119894 120573119894 fulfilling120572119894 le 1 minus 1205914120574 lt 120573119894 le 8120574

119894 = 1 2(12)

Clearly (11) holds Take a lower bound V = 0 and upper boundV is to be defined later Moreover the initial dates 11990610 and 11990620are satisfied as

1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω) le ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω) le ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1

(13)

Then consider the following

(1) Condition (6) is equivalent to 120574120572119894(V(120573119894 + V)) le 120581119894 sochoosing 120581119894 = 120574120572119894(V(120573119894 + V)) (119894 = 1 2)

(2) Taking positive constants 1205810119894 = 120572119894120573minus120572119894119894 (120573119894+V)1minus120572119894 (119894 =1 2) then condition (7) holds since

1205810119894 le 120572119894120573119894 + V119890120572119894 intV0 (1(120573119894+119904))119889119904 = 120572119894(120573119894 + V)1minus120572119894 (

1120573119894)120572119894

(14)

for 119894 = 1 2

(3) We notice that ℎ(0 0 0) = 0 as well asℎ (1199061 1199062 V) = 1199061 + 1199062 minus 120574V

= 119890(1205721(1minus120591)) intV0 (1(1205731+119904))119889119904 1205811 (1 minus 120591) + 120583112058101 + 1205831+ 119890(1205722(1minus120591)) intV0 (1(1205732+119904))119889119904 1205812 (1 minus 120591) + 120583212058102 + 1205832 minus 120574V

(15)

A sufficient condition for the second inequality in (8)holding is

119890(1205721(1minus120591)) intV0 (1(1205731+119904))119889119904 1205811 (1 minus 120591) + 120583112058101 + 1205831 lt 1205742V119890(1205722(1minus120591)) intV0 (1(1205732+119904))119889119904 1205812 (1 minus 120591) + 120583212058102 + 1205832 lt 1205742V

(16)

and for simplicity let us take 120583119894 = 120581119894(1minus120591) (119894 = 1 2)and then derive

V gt max 1205731(1205744) 1205731 minus 1 1205732(1205744) 1205732 minus 1 (17)

Up to now the above all restrictive conditions are verifiedwhich implies that conditions (6)ndash(11) are sufficient to ensurethe global existence of solutions

Remark 2 In literature [7] the chemical production functionℎ(1199061 1199062 V) = minusV + 1199061 + 1199062 and the chemotactic sensitivity120594119894(V) = 120574119894(1 + 120574119894V) (119894 = 1 2)In this paper the purpose is to study the global existence

and uniform boundedness of solutions to (1) applying aniterative methodThemain results are stated by the followingtheorem

Theorem 3 Under assumptions (6)ndash(11) for any initial date(11990610 11990620 V0) isin (1198621(Ω))3 satisfying the homogeneous Neu-mann boundary conditions and 11990610 ge 0 11990620 ge 0 there exists aunique solution to (1)

(1199061 1199062 V)isin (119871119901 ([0infin) 1198822119901 (Ω)) cap1198821119901 ([0infin) 119871119901 (Ω)))3 (18)

for any 119901 gt 119899 Moreover the solution is uniformly boundedthat is

100381710038171003817100381711990611003817100381710038171003817119871infin(Ω) + 100381710038171003817100381711990621003817100381710038171003817119871infin(Ω) + V119871infin(Ω) le 119862 (19)

where 119862 (ltinfin) is positive constantThe proof of Theorem 3 is split into several steps Step

one we start to consider the local existence of solutions andthen proceed with the extensibility criterion Step two undersome appropriate conditions we prove that the solutions areuniformly bounded in time

4 Discrete Dynamics in Nature and Society

3 Global Existence of Solutions

We will first be devoted to dealing with local-in-time exis-tence and uniqueness of a nonnegative solution for (1) Thecorresponding conclusions are written by Lemma 4

31 Local Existence of Solutions

Lemma 4 Given positive initial date (11990610 11990620 V0) isin (1198621(Ω))3and the same assumptions as inTheorem 3 there exists a smallenough time 119879 gt 0 and a unique triple of nonnegativefunctions (1199061 1199062 V) isin (119871119901([0 119879]1198822119901(Ω)) cap 1198821119901([0 119879]119871119901(Ω)))3 (for 119901 gt 119899) such that (1199061 1199062 V) is a solution of (1)in Ω times (0 119879) Moreover we have V ge V

Proof Introduce the change of variables 1 and 2 given by

1199061 = 1198911 (V) 11199062 = 1198912 (V) 2 (20)

where the function 119891119894(V) is defined by

119891119894 (V) = exp 11 minus 120591 intV

V120594119894 (119904) 119889119904 119894 = 1 2 (21)

A direct calculation yields

1199061119905 = 11989110158401 (V) (120591ΔV + ℎ (1199061 1199062 V)) 1+ 1198911 (V) 1119905

Δ1199061 = 119891101584010158401 (V) 1 |nablaV|2 + 211989110158401 (V) nabla1 sdot nablaV+ 11989110158401 (V) 1ΔV + 1198911 (V) Δ1

nabla sdot (11990611205941 (V) nablaV) = 11989110158401 (V) 11205941 (V) |nablaV|2+ 1198911 (V) 1205941 (V) nabla1 sdot nablaV+ 1198911 (V) 112059410158401 (V) |nablaV|2+ 1198911 (V) 11205941 (V) ΔV

(22)

similarly

1199062119905 = 11989110158402 (V) (120591ΔV + ℎ (1199061 1199062 V)) 2+ 1198912 (V) 2119905

Δ1199062 = 119891101584010158402 (V) 2 |nablaV|2 + 211989110158402 (V) nabla2 sdot nablaV+ 11989110158402 (V) 2ΔV + 1198912 (V) Δ2

nabla sdot (11990621205942 (V) nablaV) = 11989110158402 (V) 21205942 (V) |nablaV|2+ 1198912 (V) 1205942 (V) nabla2 sdot nablaV+ 1198912 (V) 212059410158402 (V) |nablaV|2+ 1198912 (V) 21205942 (V) ΔV

(23)

where

1198911015840119894 (V) = 120594119894 (V)1 minus 120591119891119894 (V) 11989110158401015840119894 (V) = ( 1205942119894 (V)(1 minus 120591)2 +

1205941015840119894 (V)1 minus 120591 )119891119894 (V) 119894 = 1 2

(24)

Substitute (20) (22) and (23) into (1) and denote that

L119894 (V) 119894 = 119894119905 minus Δ119894 minus 1 + 1205911 minus 120591120594119894 (V) nablaV sdot nabla119894119866119894 (1 2 V) = minus119909119894 (V)1 minus 120591 ℎ (1198911 (V) 1 1198912 (V) 2 V)+ 120583119894119892119894 (1198911 (V) 1 1198912 (V) 2)

(25)

for 119894 = 1 2Then system (1) becomes

L1 (V) 1= 11198661 (1 2 V)

+ 1 120591(1 minus 120591)2 (12059421 (V) + (1 minus 120591) 12059410158401 (V)) |nablaV|2

119909 isin Ω 119905 gt 0L2 (V) 2

= 21198662 (1 2 V)+ 2 120591

(1 minus 120591)2 (12059422 (V) + (1 minus 120591) 12059410158402 (V)) |nablaV|2 119909 isin Ω 119905 gt 0

V119905 = 120591ΔV + ℎ (1198911 (V) 1 1198912 (V) 2 V) 119909 isin Ω 119905 gt 0

(26)

with boundary conditions 1205971120597] = 1205972120597] = 120597V120597] = 0119909 isin 120597Ω 119905 gt 0 and initial conditions 1(119909 0) fl 10(119909) =11990610(119909)1198911(V0(119909)) 2(119909 0) fl 20(119909) = 11990620(119909)1198912(V0(119909))V(119909 0) = V0(119909) 119909 isin Ω

For sufficiently small 119879 gt 0 in the space [119871119901([0 119879]1198822119901(Ω)) cap 1198821119901([0 119879] 119871119901(Ω))]2 (for 119901 gt 119899) we implementa fixed point argument Taking V isin 119862([0 119879] 1198621(Ω)) satisfyingV isin (V V) and |nablaV| lt 119862 we define 1 2 in (0 119879) as the uniquesolution to the questionL1 (V) 1

= 11198661 (1 2 V)+ 1 120591

(1 minus 120591)2 (12059421 (V) + (1 minus 120591) 12059410158401 (V)) |nablaV|2in Ω times (0 119879)

L2 (V) 2= 21198662 (1 2 V)

+ 2 120591(1 minus 120591)2 (12059422 (V) + (1 minus 120591) 12059410158402 (V)) |nablaV|2

in Ω times (0 119879)

(27)

Discrete Dynamics in Nature and Society 5

for the details we refer the reader to [13 Remark 483]Noticethat the right-hand side terms of (27) are the multiplicativefor 1 and 2 applying the maximum principle to verify thatboth 1 and 2 are nonnegative

Since condition (11) we easily see that the regular func-tions 1 and 2 satisfy

L1 (V) 1 le 11198661 (1 2 V) L2 (V) 2 le 21198662 (1 2 V) (28)

and the right-hand side terms of (27) are quasi-decreasedlooking upon the lower-solution 1199061 = 1199062 = 0 one mayconstruct upper-solutions 1199061 and 1199062 such that

0 le 1199061 le 11990610 le 1199062 le 1199062119905 lt 119879

(29)

It follows that we apply Theorem 21 of [7] to get a uniquenonnegative solution

We solve the parabolic equation

V119905 = 120591ΔV + ℎ (1198911 (V) 1 1198912 (V) 2 V) (30)

Thanks to the essential estimations of parabolic equations andthe embedding theorems we apply the Schauder fixed pointtheorem to obtain the local existence of solution V(119909 119905) Thesmoothness of ℎ(1198911(V)1 1198912(V)2 V) ensures the uniquenessof solution V(119909 119905)

To prove V ge V in view of assumption (8) and themonotone of ℎ(1199061 1199062 V) we have 0 le ℎ(0 0 V) le ℎ(1199061 1199062 V)It follows that V is a lower-solution for the equation

V119905 minus 120591ΔV = ℎ (1199061 1199062 V) in Ω times (0 119879) 120597V120597] = 0 on 120597Ω times (0 119879)

V (119909 0) = V0 (119909) in Ω(31)

The proof of Lemma 4 is completed

About the above solution (1199061 1199062 V) we have the followingextensibility criterionThe solution is extended to the interval(0 119879max) where 119879max has the following property

If 119879max lt infinthen lim sup

119905rarr119879max

(10038171003817100381710038171199061 (sdot 119905)1003817100381710038171003817119871infin(Ω) + 10038171003817100381710038171199062 (sdot 119905)1003817100381710038171003817119871infin(Ω)+ V (sdot 119905)119871infin(Ω)) = infin

(32)

32 Uniform Boundedness of Solutions To obtain some a pri-ori estimates we need some technical lemmasThe following1198711-estimate of solution is first given

Lemma 5 For all 119905 isin (0 119879119898119886119909) the solutions to (1) satisfy thefollowing estimates

1003817100381710038171003817119906110038171003817100381710038171198711(Ω) le max |Ω| 10038171003817100381710038171199061010038171003817100381710038171198711(Ω) le 1198721003817100381710038171003817119906210038171003817100381710038171198711(Ω) le max |Ω| 10038171003817100381710038171199062010038171003817100381710038171198711(Ω) le 119872V1198711(Ω) le max 1003817100381710038171003817V010038171003817100381710038171198711(Ω) 2119872120574minus1

(33)

where119872 = max|Ω| 119906101198711(Ω) 119906201198711(Ω)Proof Integrating the first equation of (1) overΩ we have

119889119889119905 intΩ 1199061119889119909 = 1205831 intΩ(1199061 minus 11990621 minus 119886111990611199062) 119889119909

le 1205831 intΩ(1199061 minus 11990621) 119889119909

(34)

Using Cauchy inequality yields

119889119889119905 intΩ 1199061119889119909 le 1205831 intΩ(1 minus 1199061) 119889119909

le 1205831 |Ω| minus 1205831 intΩ1199061119889119909

(35)

By the Gronwall Lemma we derive

10038171003817100381710038171199061 (119909 119905)10038171003817100381710038171198711(Ω) le max |Ω| 100381710038171003817100381711990610 (119909)10038171003817100381710038171198711(Ω) le 119872 (36)

a similar estimate holds for 1199062 which leads to

10038171003817100381710038171199062 (119909 119905)10038171003817100381710038171198711(Ω) le max |Ω| 100381710038171003817100381711990620 (119909)10038171003817100381710038171198711(Ω) le 119872 (37)

Integrating the third equation of (1) overΩ we have

119889119889119905 intΩ V 119889119909 = intΩ(minus120574V + 1199061 + 1199062) 119889119909

= minus120574intΩV 119889119909 + int

Ω(1199061 + 1199062) 119889119909

le minus120574intΩV 119889119909 + 2119872

(38)

it implies

V1198711(Ω) le max 1003817100381710038171003817V010038171003817100381710038171198711(Ω) 2119872120574minus1 (39)

This proves the lemma

6 Discrete Dynamics in Nature and Society

Lemma 6 Letting 119901 gt 1 and under assumption (11) then thefollowing estimates hold

119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909le 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990111205941 (V) 1198911minus1199011119901 ℎ (1199061 1199062 V) 119889119909+ 1199011205831 int

Ω11990611990111198911minus1199011119901 1198921 (1199061 1199062) 119889119909

(40)

119889119889119905 intΩ 11990611990121198911minus1199012119901 119889119909le 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990121205942 (V) 1198911minus1199012119901 ℎ (1199061 1199062 V) 119889119909+ 1199011205832 int

Ω11990611990121198911minus1199012119901 1198922 (1199061 1199062) 119889119909

(41)

where 119891119894119901(V) is defined by

119891119894119901 (V) = exp( 11 minus 120591 ((119901 minus 1) 119901) intV

V120594119894 (119904) 119889119904)

119894 = 1 2(42)

Proof It is easy to check

11989110158401119901 (V) = 11 minus 120591 ((119901 minus 1) 119901)1205941 (V) 1198911119901 (V) 11989110158402119901 (V) = 11 minus 120591 ((119901 minus 1) 119901)1205942 (V) 1198912119901 (V)

(43)

Then for any 119901 gt 1 we have119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909 = 119901int

Ω119906119901minus11 11990611199051198911minus1199011119901 119889119909 + (1 minus 119901)

sdot intΩ1199061199011119891minus1199011119901 11989110158401119901V119905119889119909 = 119901int

Ω119906119901minus11 1198911minus1199011119901 [Δ1199061 minus nabla

sdot (11990611205941 (V) nablaV) + 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1)11990611205941 (V) ΔV]119889119909

+ 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990111198911minus1199011119901 1205941 (V)

sdot ℎ (1199061 1199062 V) 119889119909+ 1199011205831 int

Ω11990611990111198911minus1199011119901 1198921 (1199061 1199062) 119889119909 fl 1198681 + 1198682 + 1198683

(44)

Clearly if 1198681 le 0 then (40) is a consequence of (44) We inthe following verify that the result holds

In fact

1198681 = 119901intΩ119906119901minus11 1198911minus1199011119901 [Δ1199061 minus nabla sdot (11990611205941 (V) nablaV)

+ 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1) (nabla sdot (11990611205941 (V) nablaV) minus nabla (11990611205941 (V))

sdot nablaV)] 119889119909 = 119901intΩ119906119901minus11 1198911minus1199011119901 [nabla

sdot (1198911119901nabla (1199061119891minus11119901 )) minus 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1) (1205941 (V) nabla1199061 sdot nablaV

+ 119906112059410158401 (V) |nablaV|2)] 119889119909 = 119901 (1 minus 119901)

sdot intΩ119906119901minus21 1198912minus1199011119901 [1198911119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162

minus 1205911199061 (1205941 (V) nabla1199061 sdot nablaV + 119906112059410158401 (V) |nablaV|2)1198911119901 (119901 minus 120591 (119901 minus 1)) ] 119889119909= 119901 (1 minus 119901)int

Ω119906119901minus21 1198913minus1199011119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162 119889119909

+ 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ 119906119901minus11 1198911minus1199011119901 1205941 (V) nabla1199061 sdot nablaV 119889119909

+ 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ 11990611990111198911minus1199011119901 12059410158401 (V) |nablaV|2 119889119909 fl 1198691

+ 1198692 + 1198693

(45)

Since nabla1199061 = nabla(1198911119901(1199061119891minus11119901 )) multiplying both sides by1205941(V)nablaV we get1205941 (V) nabla1199061 sdot nablaV

= 11989111199011205941 (V) nabla (1199061119891minus11119901 ) sdot nablaV+ 11 minus 120591 ((119901 minus 1) 119901)119906112059421 (V) |nablaV|2

(46)

and by 120576-Youngrsquos inequality1205941 (V) nabla (1199061119891minus11119901 ) sdot nablaV

le 119901 minus 120591 (119901 minus 1)120591

1198911119901119906110038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162

+ 120591119901 minus 120591 (119901 minus 1) 1199061119891111990112059421 (V) |nablaV|2

(47)

Combining (46) and (47) and substituting it into 1198692 we derive1198692 le 119901 (119901 minus 1)int

Ω119906119901minus21 1198913minus1199011119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162 119889119909

+ 120591119901 (119901 minus 1) (120591 + 119901)(119901 minus 120591 (119901 minus 1))2 int

Ω11990611990111198911minus1199011119901 12059421 (V) |nablaV|2 119889119909

(48)

Discrete Dynamics in Nature and Society 7

Inserting (48) into (45) yields

1198681 le 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ (

120591 + 119901(119901 minus 120591 (119901 minus 1))12059421 (V)

+ 12059410158401 (V)) 11990611990111198911minus1199011119901 |nablaV|2 119889119909 = 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1)

sdot intΩ[ 1(1 minus 120591) (1 minus 1205912120591 + (1 minus 120591) 119901)12059421 (V)

+ 12059410158401 (V)] 11990611990111198911minus1199011119901 |nablaV|2 119889119909 le 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1)

sdot intΩ( 1(1 minus 120591)12059421 (V) + 12059410158401 (V)) 11990611990111198911minus1199011119901 |nablaV|2 119889119909

(49)

we deduce from assumption (11) that 1198681 le 0 for any 119901 gt 1and the proof ends In the same way we achieve (41) Thiscompletes the proof of Lemma 6

In view of V isin 119862((0 119879max) 1198621(Ω)) and initial date V ltV0 lt V we consider 119879lowast gt 0 defined by

119879lowast fl 119879max if V le V in (0 119879max) 119879V if V119871infin(Ω119879max )

gt V (50)

where 119879V lt 119879max satisfies that V119871infin(Ω119879V ) le V andV119871infin(Ω119879V+120575) gt V for any 120575 gt 0Lemma7 Letting119901 gt 1 and for any 119905 lt 119879lowast then the solutionsto (1) satisfy

1119901 minus 1 119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909le minus (12058101 + 1205831) int

Ω119906119901+11 119891minus1199011119901 119889119909

+ ( 12058111 minus 120591 + 119901119901 minus 11205831)intΩ11990611990111198911minus1199011119901 119889119909

(51)

1119901 minus 1 119889119889119905 intΩ 11990611990121198911minus1199012119901 119889119909le minus (12058102 + 1205832) int

Ω119906119901+12 119891minus1199012119901 119889119909

+ ( 12058121 minus 120591 + 119901119901 minus 11205832)intΩ11990611990121198911minus1199012119901 119889119909

(52)

Proof In order to prove Lemma 7 we consider the terms 1198682and 1198683 appearing in the proof of Lemma 6

By the nonnegativity for 1199061 1199062 and assumption (6) weobtain

minusℎ (1199061 1199062 V) le minus1199061 + 12058111205941 (V) (53)

Multiplying (53) by 11990611990111198911minus1199011119901 1205941(V) and integrating overΩ yield

minus intΩ11990611990111198911minus1199011119901 1205941 (V) ℎ (1199061 1199062 V) 119889119909

le minusintΩ119906119901+11 1198911minus1199011119901 1205941 (V) 119889119909 + 1205811 int

Ω11990611990111198911minus1199011119901 119889119909

(54)

By the monotonic property of exponential function andassumption (7) we have

0 lt 12058101 le 1205941 (V) exp(intV

V1205941 (119904) 119889119904) le 1205941 (V) 1198911119901 (V) (55)

It follows that (54) becomes

minus intΩ11990611990111198911minus1199011119901 1205941 (V) ℎ (1199061 1199062 V) 119889119909

le minus12058101 intΩ119906119901+11 119891minus1199011119901 119889119909 + 1205811 int

Ω11990611990111198911minus1199011119901 119889119909

(56)

Multiplying (56) by (119901 minus 1)(1 minus 120591((119901 minus 1)119901)) we get1198682 le 119901 minus 1

1 minus 120591 ((119901 minus 1) 119901) (minus12058101 intΩ 119906119901+11 119891minus1199011119901 119889119909+ 1205811 int

Ω11990611990111198911minus1199011119901 119889119909) le (119901 minus 1)

sdot (minus12058101 intΩ119906119901+11 119891minus1199011119901 119889119909 + 12058111 minus 120591 int

Ω11990611990111198911minus1199011119901 119889119909)

(57)

For the term 1198683 we easily achieve1198683 le 1199011205831 (int

Ω11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

le (119901 minus 1) 1205831 (intΩ11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

+ 1205831 intΩ11990611990111198911minus1199011119901 119889119909

(58)

Combining (44) (57) and (58) leads to

119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909 le (119901 minus 1)sdot (minus12058101 int

Ω119906119901+11 119891minus1199011119901 119889119909 + 12058111 minus 120591 int

Ω11990611990111198911minus1199011119901 119889119909)

+ (119901 minus 1) 1205831 (intΩ11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

+ 1205831 intΩ11990611990111198911minus1199011119901 119889119909

(59)

which implies that (51) holds The same arguments are usedto achieve (52) The proof ends

To establish the uniform boundedness of solutions weapply an iterative method based on the 119871119901-norm of solutionsinspired by the Moser-Alikakos type iterations (see [10 13])

8 Discrete Dynamics in Nature and Society

Lemma 8 For any 119905 lt 119879lowast then the solutions to (1) arebounded by

100381710038171003817100381711990611003817100381710038171003817119871infin(Ω) le 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

(60)

100381710038171003817100381711990621003817100381710038171003817119871infin(Ω) le 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(61)

Proof DenoteintΩ 11990611990111198911minus1199011119901 119889119909 byΨ(119905) Taking arbitrary positiveconstant 120588 gt 0 then it holds that

Ψ (119905) = int1198911119901le120588(12058101+1205831)1199061

11990611990111198911minus1199011119901 119889119909+ int1198911119901gt120588(12058101+1205831)1199061

11990611990111198911minus1199011119901 119889119909le 120588 (12058101 + 1205831) int

1198911119901le120588(12058101+1205831)1199061119906119901+11 119891minus1199011119901 119889119909

+ (120588 (12058101 + 1205831))1minus119901 int1198911119901gt120588(12058101+1205831)1199061

1199061119889119909le 120588 (12058101 + 1205831) int

Ω119906119901+11 119891minus1199011119901 119889119909

+ (120588 (12058101 + 1205831))1minus119901 intΩ1199061119889119909

(62)

Hence

minus (12058101 + 1205831) intΩ119906119901+11 119891minus1199011119901 119889119909

le minus1120588Ψ (119905) + 120588minus119901 (12058101 + 1205831)1minus119901 intΩ1199061119889119909

(63)

Substituting (63) into (51) we have

1119901 minus 1 119889119889119909Ψ (119905) le ( 12058111 minus 120591 + 119901119901 minus 11205831 minus 1120588)Ψ (119905)+ 120588minus119901 (12058101 + 1205831)1minus119901 int

Ω1199061119889119909

(64)

Take 1120588 gt 1205811(1 minus 120591) + (119901(119901 minus 1))1205831 Noticing Lemma 5and initial date 11990610 isin 1198621(Ω) we apply the Gronwall Lemmato obtain

Ψ (119905) le max120588minus119901 (12058101 + 1205831)1minus119901

sdot ( 1120588 minus 12058111 minus 120591 minus 119901119901 minus 11205831)minus1 intΩ1199061119889119909 Ψ (0)

(65)

taking limits leads to

lim119901rarrinfin

Ψ1119901 (119905) le max 120588minus1 (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω) (66)

Since

Ψ (119905) = intΩ11990611990111198911minus1199011119901 (V) 119889119909 ge 1198911minus1199011119901 (V) int

Ω1199061199011119889119909 (67)

combining (66) and (67) and taking limits as 120588minus1 rarr 1205811(1 minus120591) + 1205831 it is found that (60) holds The same processes areused to prove (61) which ends the proof of Lemma 8

Remark 9 In the above arguments we used well such factLetΩ be bounded and regular domain ofR119899 If 119906119871119901(Ω) le 119862where119862 is some positive constant and independent of119901 then119906119871119901(Ω) rarr 119906119871infin(Ω) as 119901 rarr infin

To end the arguments of uniform boundedness of solu-tions it follows to prove that 119879lowast = 119879max and 119879max = +infin bycontradiction

As in Lemmas 4ndash8 we define 1 2 and V for the uniquesolutions in (0 max) to

1119905 = Δ1 minus nabla sdot (11205941 (V) nablaV) + 120583111198921 (1 2) 119909 isin Ω 119905 gt 0

2119905 = Δ2 minus nabla sdot (21205942 (V) nablaV) + 120583211990621198922 (1 2) 119909 isin Ω 119905 gt 0

V119905 = 120591ΔV + ℎ (1 2 V) 119909 isin Ω 119905 gt 01205971120597] = 1205972120597] = 120597V120597] = 0 119909 isin 120597Ω 119905 gt 0

1 (119909 0) = 11990610 (119909) 2 (119909 0) = 11990620 (119909) V (119909 0) = V0 (119909)

119909 isin Ω

(68)

where max is defined in the same fashion as (32) and thetruncation ℎ(1199061 1199062 V) is defined as follows

ℎ (1199061 1199062 V) fl ℎ (1199061 1199062 V) if V le Vℎ (1199061 1199062 V) if V gt V (69)

Then it is easy to see that100381710038171003817100381711003817100381710038171003817119871infin(Ω) le 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

100381710038171003817100381721003817100381710038171003817119871infin(Ω) le 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(70)

as far as V le VBy analogy with 119879lowast we define

lowast fl max if V le V in (0 max) V if V119871infin(Ωmax )

gt V (71)

Discrete Dynamics in Nature and Society 9

where V lt max satisfies that V119871infin(ΩV ) le V and

V119871infin(ΩV+120575) gt V for any 120575 gt 0 To prove lowast = max wesuppose

lowast lt max (72)

then ℎ(1 2 V) le ℎ(1199061 1199062 V) = ℎ(1199061 1199062 V) lt 0 It follows toeasily check that V is a strict upper-solution to

V119905 minus 120591ΔV = ℎ (1 2 V) in Ω times (0 lowast) 120597V120597] = 0 on 120597Ω times (0 lowast)

V (119909 0) = V0 (119909) in Ω(73)

This implies V(119909 119905) lt V forall119909 isin Ω 119905 isin (0 lowast) whichcontradicts (72) and proves lowast = max as well as max = infin

Thanks to V le V we show that (1 2 V) is also a solutionto (6) So we have

119879lowast = 119879max ge max = infin (74)

The proof of Theorem 3 is completed

Competing Interests

The author declares that they have no competing interests

Acknowledgments

Thiswork is supported by the ChinaNational Natural ScienceFoundation (nos 11361055 and 11261053) and the NaturalScience ResearchKey Project of University of Anhui Province(no KJ2015A251)

References

[1] J D Murray Mathematical Biology vol 19 Springer BerlinGermany 2nd edition 1993

[2] M Winkler ldquoBoundedness in the higher-dimensionalparabolic-parabolic chemotaxis system with logistic sourcerdquoCommunications in Partial Differential Equations vol 35 no 8pp 1516ndash1537 2010

[3] T Hillen and K J Painter ldquoA userrsquos guide to PDE models forchemotaxisrdquo Journal of Mathematical Biology vol 58 no 1-2pp 183ndash217 2009

[4] E F Keller and L A Segel ldquoInitiation of slimemold aggregationviewed as an instabilityrdquo Journal of Theoretical Biology vol 26no 3 pp 399ndash415 1970

[5] M Winkler ldquoAbsence of collapse in a parabolic chemo-taxis system with signal-dependent sensitivityrdquo MathematischeNachrichten vol 283 no 11 pp 1664ndash1673 2010

[6] J I Tello and M Winkler ldquoStabilization in a two-specieschemotaxis system with a logistic sourcerdquo Nonlinearity vol 25no 5 pp 1413ndash1425 2012

[7] M Negreanu and J I Tello ldquoAsymptotic stability of a twospecies chemotaxis systemwith non-diffusive chemoattractantrdquoJournal of Differential Equations vol 258 no 5 pp 1592ndash16172015

[8] Q Zhang and Y Li ldquoGlobal boundedness of solutions toa two-species chemotaxis systemrdquo Zeitschrift fur angewandteMathematik und Physik vol 66 no 1 pp 83ndash93 2015

[9] C Mu L Wang P Zheng and Q Zhang ldquoGlobal existenceand boundedness of classical solutions to a parabolic-parabolicchemotaxis systemrdquo Nonlinear Analysis Real World Applica-tions vol 14 no 3 pp 1634ndash1642 2013

[10] Y Tao ldquoBoundedness in a chemotaxis model with oxygenconsumption by bacteriardquo Journal ofMathematical Analysis andApplications vol 381 no 2 pp 521ndash529 2011

[11] D Horstmann and M Winkler ldquoBoundedness vs blow-up in achemotaxis systemrdquo Journal of Differential Equations vol 215no 1 pp 52ndash107 2005

[12] M Winkler ldquoAggregation vs global diffusive behavior in thehigher-dimensional Keller-Segel modelrdquo Journal of DifferentialEquations vol 248 no 12 pp 2889ndash2905 2010

[13] PQuittner andP Souplet Superlinear Parabolic Problems Blow-Up Global Existence and Steady States Birkhauser AdvancedTexts Birkhauser Basel Switzerland 2007

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2 Discrete Dynamics in Nature and Society

equations to describe the aggregation of Dictyostelium dis-coideum as well as a soil-living amoeba in the early 1970s [4]After the pioneeringworks of Keller and Segel a large amountof chemotaxismodels has been used tomodel the phenomenafor population dynamics or gravitational collapse amongothers Winkler [5] studied the chemotaxis system

119906119905 = Δ119906 minus nabla sdot (119906120594 (V) nablaV) 119909 isin Ω 119905 gt 0V119905 = ΔV minus V + 119906 119909 isin Ω 119905 gt 0 (2)

under homogeneous Neumann boundary conditions in asmooth bounded domain Ω sub R119899 It was proved that thechemotactic collapse was absented for any nonnegative initialdate 119906(sdot 0) isin 1198620(Ω) and V(sdot 0) isin 1198821119903(Ω) with some119903 gt 119899 the corresponding initial-boundary value problempossessed a unique global uniformly bounded solution Telloand Winkler [6] considered the parabolic-parabolic-ellipticsystem

119906119905 = Δ119906 minus 1205941nabla sdot (119906nabla119908) + 1205831119906 (1 minus 119906 minus 1198861V) 119909 isin Ω 119905 gt 0

V119905 = ΔV minus 1205942nabla sdot (Vnabla119908) + 1205832V (1 minus 1198862119906 minus V) 119909 isin Ω 119905 gt 0

minusΔ119908 = minus120582119908 + 119906 + V 119909 isin Ω 119905 gt 0

(3)

under homogeneous Neumann boundary conditions in abounded domain Ω sub R119899 (119899 ge 1) with smooth boundaryGiven the suitable positive parameters 1198861 1198862 and 1205941 1205942 theyshowed that all solutions stabilized towards a uniquely deter-mined spatially homogeneous positive steady state within acertain nonempty range of the logistic growth coefficients1205831 and 1205832 Negreanu and Tello [7] studied a two-specieschemotaxis system with nondiffusive chemoattractant

119906119905 = Δ119906 minus nabla sdot (1199061205941 (119908) nabla119908) + 1205831119906 (1 minus 119906) 119909 isin Ω 119905 gt 0

V119905 = ΔV minus nabla sdot (V1205942 (119908) nabla119908) + 1205832V (1 minus V) 119909 isin Ω 119905 gt 0

119908119905 = ℎ (119906 V 119908) 119909 isin Ω 119905 gt 0

(4)

under suitable boundary and initial conditions in an 119899-dimensional open and bounded domain Ω for 119899 ge 1 Theyconsidered the case of positive chemosensitivities and chem-ical production function ℎ increasing as the concentrationof the species 119906 V increasing The paper proved the globalexistence and uniform boundedness of solutions and theasymptotic stability of the spatially homogeneous steady statewas a consequence of the growth of ℎ 120594119894 and the size of 120583119894 for119894 = 1 2

Reviewing the recent studies Zhang andLi [8] consideredthe following fully parabolic system

119906119905 = Δ119906 minus nabla sdot (1199061205941 (119908) nabla119908) + 1205831119906 (1 minus 119906 minus 1198861V) 119909 isin Ω 119905 gt 0

V119905 = ΔV minus nabla sdot (V1205942 (119908) nabla119908) + 1205832V (1 minus 1198862119906 minus V) 119909 isin Ω 119905 gt 0

119908119905 = Δ119908 minus 119908 + 119906 + V 119909 isin Ω 119905 gt 0

(5)

under homogeneous Neumann boundary conditions in asmooth bounded domain Ω sub R119899 (119899 ge 1) By extendingthe method in [5] (see also [9 10]) the first step is to esti-mate some associated weighted functions which depend onsignal density and the second step is to obtain 119871infin-boundsof solutions from 119871119901-bounds using the variation-of-constantsrepresentation and a series of standard semigroup arguments(see [2 5 11 12]) They proved that if the nonnegative initialdate (119906(sdot 0) V(sdot 0)) isin (1198620(Ω))2 and 119908(sdot 0) isin 1198821119903(Ω) forsome 119903 gt 119899 the systempossesses a unique global solution thatis uniformly bounded under some appropriate conditions onthe coefficients 1205831 1205832 and the chemotaxis sensitivity func-tions 1205941(119908) 1205942(119908)

Inspired by the foregoing research the main purposeof this paper is to consider the existence of global solutionfor the two-species competitive chemotaxis model (1) withinequal diffusion rates This paper is organized as followsIn Section 2 we formulate the main results of this paper bymeans of the theorem and establish some preliminarieswhichare important for our proofs In Section 3 we firstly considerthe local existence of solutions and then proceed with theextensibility criterion Finally under some appropriate con-ditions we prove that the solutions are uniformly boundedin time using an iterative method

2 Preliminaries and Main Results

For convenience we denote that 1198921(1199061 1199062) = 1 minus 1199061 minus 119886111990621198922(1199061 1199062) = 1 minus 11988621199061 minus 1199062 and they are quasi-monotonicdecrease in R2+ Denote ℎ(1199061 1199062 V) = minus120574V + 1199061 + 1199062

In order to establish the global existence and uniformboundedness of solutions to (1) we need to make somerestrictive conditions throughout this paper in ℎ(1199061 1199062 V)and 120594119894(V) (119894 = 1 2)

(i) Let the initial date V0 satisfy 0 le V lt V0 lt V where Vand V are some positive constants

(ii) There exist positive constants 1205811 and 1205812 such that

120574V120594119894 (V) le 120581119894 for V le V le V 119894 = 1 2 (6)

(iii) There exist positive constants 12058101 and 12058102 such that

0 lt 1205810119894 le 120594119894 (V) exp(intV

V120594119894 (119904) 119889119904)

for V le V 119894 = 1 2(7)

Discrete Dynamics in Nature and Society 3

(iv) Assume that

ℎ (1199061 1199062 V) ge 0ℎ (1199061 1199062 V) lt 0

for 0 le 1199061 le 1199061 0 le 1199062 le 1199062(8)

where

1199061 fl 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

1199062 fl 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(9)

for 119891119894(V) defined by

119891119894 (V) = exp( 11 minus 120591 intV

V120594119894 (119904) 119889119904) 119894 = 1 2 (10)

(v) Finally for some technical reasons we also assumethat

1205942119894 (V) + (1 minus 120591) 1205941015840119894 (V) le 0 119894 = 1 2 (11)

We illustrate the validity to above assumptions with thefollowing generalized example [7]

Example 1 We take the chemosensitivity functions 120594119894(V) =120572119894(120573119894 + V) (119894 = 1 2) for positive constants 120572119894 120573119894 fulfilling120572119894 le 1 minus 1205914120574 lt 120573119894 le 8120574

119894 = 1 2(12)

Clearly (11) holds Take a lower bound V = 0 and upper boundV is to be defined later Moreover the initial dates 11990610 and 11990620are satisfied as

1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω) le ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω) le ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1

(13)

Then consider the following

(1) Condition (6) is equivalent to 120574120572119894(V(120573119894 + V)) le 120581119894 sochoosing 120581119894 = 120574120572119894(V(120573119894 + V)) (119894 = 1 2)

(2) Taking positive constants 1205810119894 = 120572119894120573minus120572119894119894 (120573119894+V)1minus120572119894 (119894 =1 2) then condition (7) holds since

1205810119894 le 120572119894120573119894 + V119890120572119894 intV0 (1(120573119894+119904))119889119904 = 120572119894(120573119894 + V)1minus120572119894 (

1120573119894)120572119894

(14)

for 119894 = 1 2

(3) We notice that ℎ(0 0 0) = 0 as well asℎ (1199061 1199062 V) = 1199061 + 1199062 minus 120574V

= 119890(1205721(1minus120591)) intV0 (1(1205731+119904))119889119904 1205811 (1 minus 120591) + 120583112058101 + 1205831+ 119890(1205722(1minus120591)) intV0 (1(1205732+119904))119889119904 1205812 (1 minus 120591) + 120583212058102 + 1205832 minus 120574V

(15)

A sufficient condition for the second inequality in (8)holding is

119890(1205721(1minus120591)) intV0 (1(1205731+119904))119889119904 1205811 (1 minus 120591) + 120583112058101 + 1205831 lt 1205742V119890(1205722(1minus120591)) intV0 (1(1205732+119904))119889119904 1205812 (1 minus 120591) + 120583212058102 + 1205832 lt 1205742V

(16)

and for simplicity let us take 120583119894 = 120581119894(1minus120591) (119894 = 1 2)and then derive

V gt max 1205731(1205744) 1205731 minus 1 1205732(1205744) 1205732 minus 1 (17)

Up to now the above all restrictive conditions are verifiedwhich implies that conditions (6)ndash(11) are sufficient to ensurethe global existence of solutions

Remark 2 In literature [7] the chemical production functionℎ(1199061 1199062 V) = minusV + 1199061 + 1199062 and the chemotactic sensitivity120594119894(V) = 120574119894(1 + 120574119894V) (119894 = 1 2)In this paper the purpose is to study the global existence

and uniform boundedness of solutions to (1) applying aniterative methodThemain results are stated by the followingtheorem

Theorem 3 Under assumptions (6)ndash(11) for any initial date(11990610 11990620 V0) isin (1198621(Ω))3 satisfying the homogeneous Neu-mann boundary conditions and 11990610 ge 0 11990620 ge 0 there exists aunique solution to (1)

(1199061 1199062 V)isin (119871119901 ([0infin) 1198822119901 (Ω)) cap1198821119901 ([0infin) 119871119901 (Ω)))3 (18)

for any 119901 gt 119899 Moreover the solution is uniformly boundedthat is

100381710038171003817100381711990611003817100381710038171003817119871infin(Ω) + 100381710038171003817100381711990621003817100381710038171003817119871infin(Ω) + V119871infin(Ω) le 119862 (19)

where 119862 (ltinfin) is positive constantThe proof of Theorem 3 is split into several steps Step

one we start to consider the local existence of solutions andthen proceed with the extensibility criterion Step two undersome appropriate conditions we prove that the solutions areuniformly bounded in time

4 Discrete Dynamics in Nature and Society

3 Global Existence of Solutions

We will first be devoted to dealing with local-in-time exis-tence and uniqueness of a nonnegative solution for (1) Thecorresponding conclusions are written by Lemma 4

31 Local Existence of Solutions

Lemma 4 Given positive initial date (11990610 11990620 V0) isin (1198621(Ω))3and the same assumptions as inTheorem 3 there exists a smallenough time 119879 gt 0 and a unique triple of nonnegativefunctions (1199061 1199062 V) isin (119871119901([0 119879]1198822119901(Ω)) cap 1198821119901([0 119879]119871119901(Ω)))3 (for 119901 gt 119899) such that (1199061 1199062 V) is a solution of (1)in Ω times (0 119879) Moreover we have V ge V

Proof Introduce the change of variables 1 and 2 given by

1199061 = 1198911 (V) 11199062 = 1198912 (V) 2 (20)

where the function 119891119894(V) is defined by

119891119894 (V) = exp 11 minus 120591 intV

V120594119894 (119904) 119889119904 119894 = 1 2 (21)

A direct calculation yields

1199061119905 = 11989110158401 (V) (120591ΔV + ℎ (1199061 1199062 V)) 1+ 1198911 (V) 1119905

Δ1199061 = 119891101584010158401 (V) 1 |nablaV|2 + 211989110158401 (V) nabla1 sdot nablaV+ 11989110158401 (V) 1ΔV + 1198911 (V) Δ1

nabla sdot (11990611205941 (V) nablaV) = 11989110158401 (V) 11205941 (V) |nablaV|2+ 1198911 (V) 1205941 (V) nabla1 sdot nablaV+ 1198911 (V) 112059410158401 (V) |nablaV|2+ 1198911 (V) 11205941 (V) ΔV

(22)

similarly

1199062119905 = 11989110158402 (V) (120591ΔV + ℎ (1199061 1199062 V)) 2+ 1198912 (V) 2119905

Δ1199062 = 119891101584010158402 (V) 2 |nablaV|2 + 211989110158402 (V) nabla2 sdot nablaV+ 11989110158402 (V) 2ΔV + 1198912 (V) Δ2

nabla sdot (11990621205942 (V) nablaV) = 11989110158402 (V) 21205942 (V) |nablaV|2+ 1198912 (V) 1205942 (V) nabla2 sdot nablaV+ 1198912 (V) 212059410158402 (V) |nablaV|2+ 1198912 (V) 21205942 (V) ΔV

(23)

where

1198911015840119894 (V) = 120594119894 (V)1 minus 120591119891119894 (V) 11989110158401015840119894 (V) = ( 1205942119894 (V)(1 minus 120591)2 +

1205941015840119894 (V)1 minus 120591 )119891119894 (V) 119894 = 1 2

(24)

Substitute (20) (22) and (23) into (1) and denote that

L119894 (V) 119894 = 119894119905 minus Δ119894 minus 1 + 1205911 minus 120591120594119894 (V) nablaV sdot nabla119894119866119894 (1 2 V) = minus119909119894 (V)1 minus 120591 ℎ (1198911 (V) 1 1198912 (V) 2 V)+ 120583119894119892119894 (1198911 (V) 1 1198912 (V) 2)

(25)

for 119894 = 1 2Then system (1) becomes

L1 (V) 1= 11198661 (1 2 V)

+ 1 120591(1 minus 120591)2 (12059421 (V) + (1 minus 120591) 12059410158401 (V)) |nablaV|2

119909 isin Ω 119905 gt 0L2 (V) 2

= 21198662 (1 2 V)+ 2 120591

(1 minus 120591)2 (12059422 (V) + (1 minus 120591) 12059410158402 (V)) |nablaV|2 119909 isin Ω 119905 gt 0

V119905 = 120591ΔV + ℎ (1198911 (V) 1 1198912 (V) 2 V) 119909 isin Ω 119905 gt 0

(26)

with boundary conditions 1205971120597] = 1205972120597] = 120597V120597] = 0119909 isin 120597Ω 119905 gt 0 and initial conditions 1(119909 0) fl 10(119909) =11990610(119909)1198911(V0(119909)) 2(119909 0) fl 20(119909) = 11990620(119909)1198912(V0(119909))V(119909 0) = V0(119909) 119909 isin Ω

For sufficiently small 119879 gt 0 in the space [119871119901([0 119879]1198822119901(Ω)) cap 1198821119901([0 119879] 119871119901(Ω))]2 (for 119901 gt 119899) we implementa fixed point argument Taking V isin 119862([0 119879] 1198621(Ω)) satisfyingV isin (V V) and |nablaV| lt 119862 we define 1 2 in (0 119879) as the uniquesolution to the questionL1 (V) 1

= 11198661 (1 2 V)+ 1 120591

(1 minus 120591)2 (12059421 (V) + (1 minus 120591) 12059410158401 (V)) |nablaV|2in Ω times (0 119879)

L2 (V) 2= 21198662 (1 2 V)

+ 2 120591(1 minus 120591)2 (12059422 (V) + (1 minus 120591) 12059410158402 (V)) |nablaV|2

in Ω times (0 119879)

(27)

Discrete Dynamics in Nature and Society 5

for the details we refer the reader to [13 Remark 483]Noticethat the right-hand side terms of (27) are the multiplicativefor 1 and 2 applying the maximum principle to verify thatboth 1 and 2 are nonnegative

Since condition (11) we easily see that the regular func-tions 1 and 2 satisfy

L1 (V) 1 le 11198661 (1 2 V) L2 (V) 2 le 21198662 (1 2 V) (28)

and the right-hand side terms of (27) are quasi-decreasedlooking upon the lower-solution 1199061 = 1199062 = 0 one mayconstruct upper-solutions 1199061 and 1199062 such that

0 le 1199061 le 11990610 le 1199062 le 1199062119905 lt 119879

(29)

It follows that we apply Theorem 21 of [7] to get a uniquenonnegative solution

We solve the parabolic equation

V119905 = 120591ΔV + ℎ (1198911 (V) 1 1198912 (V) 2 V) (30)

Thanks to the essential estimations of parabolic equations andthe embedding theorems we apply the Schauder fixed pointtheorem to obtain the local existence of solution V(119909 119905) Thesmoothness of ℎ(1198911(V)1 1198912(V)2 V) ensures the uniquenessof solution V(119909 119905)

To prove V ge V in view of assumption (8) and themonotone of ℎ(1199061 1199062 V) we have 0 le ℎ(0 0 V) le ℎ(1199061 1199062 V)It follows that V is a lower-solution for the equation

V119905 minus 120591ΔV = ℎ (1199061 1199062 V) in Ω times (0 119879) 120597V120597] = 0 on 120597Ω times (0 119879)

V (119909 0) = V0 (119909) in Ω(31)

The proof of Lemma 4 is completed

About the above solution (1199061 1199062 V) we have the followingextensibility criterionThe solution is extended to the interval(0 119879max) where 119879max has the following property

If 119879max lt infinthen lim sup

119905rarr119879max

(10038171003817100381710038171199061 (sdot 119905)1003817100381710038171003817119871infin(Ω) + 10038171003817100381710038171199062 (sdot 119905)1003817100381710038171003817119871infin(Ω)+ V (sdot 119905)119871infin(Ω)) = infin

(32)

32 Uniform Boundedness of Solutions To obtain some a pri-ori estimates we need some technical lemmasThe following1198711-estimate of solution is first given

Lemma 5 For all 119905 isin (0 119879119898119886119909) the solutions to (1) satisfy thefollowing estimates

1003817100381710038171003817119906110038171003817100381710038171198711(Ω) le max |Ω| 10038171003817100381710038171199061010038171003817100381710038171198711(Ω) le 1198721003817100381710038171003817119906210038171003817100381710038171198711(Ω) le max |Ω| 10038171003817100381710038171199062010038171003817100381710038171198711(Ω) le 119872V1198711(Ω) le max 1003817100381710038171003817V010038171003817100381710038171198711(Ω) 2119872120574minus1

(33)

where119872 = max|Ω| 119906101198711(Ω) 119906201198711(Ω)Proof Integrating the first equation of (1) overΩ we have

119889119889119905 intΩ 1199061119889119909 = 1205831 intΩ(1199061 minus 11990621 minus 119886111990611199062) 119889119909

le 1205831 intΩ(1199061 minus 11990621) 119889119909

(34)

Using Cauchy inequality yields

119889119889119905 intΩ 1199061119889119909 le 1205831 intΩ(1 minus 1199061) 119889119909

le 1205831 |Ω| minus 1205831 intΩ1199061119889119909

(35)

By the Gronwall Lemma we derive

10038171003817100381710038171199061 (119909 119905)10038171003817100381710038171198711(Ω) le max |Ω| 100381710038171003817100381711990610 (119909)10038171003817100381710038171198711(Ω) le 119872 (36)

a similar estimate holds for 1199062 which leads to

10038171003817100381710038171199062 (119909 119905)10038171003817100381710038171198711(Ω) le max |Ω| 100381710038171003817100381711990620 (119909)10038171003817100381710038171198711(Ω) le 119872 (37)

Integrating the third equation of (1) overΩ we have

119889119889119905 intΩ V 119889119909 = intΩ(minus120574V + 1199061 + 1199062) 119889119909

= minus120574intΩV 119889119909 + int

Ω(1199061 + 1199062) 119889119909

le minus120574intΩV 119889119909 + 2119872

(38)

it implies

V1198711(Ω) le max 1003817100381710038171003817V010038171003817100381710038171198711(Ω) 2119872120574minus1 (39)

This proves the lemma

6 Discrete Dynamics in Nature and Society

Lemma 6 Letting 119901 gt 1 and under assumption (11) then thefollowing estimates hold

119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909le 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990111205941 (V) 1198911minus1199011119901 ℎ (1199061 1199062 V) 119889119909+ 1199011205831 int

Ω11990611990111198911minus1199011119901 1198921 (1199061 1199062) 119889119909

(40)

119889119889119905 intΩ 11990611990121198911minus1199012119901 119889119909le 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990121205942 (V) 1198911minus1199012119901 ℎ (1199061 1199062 V) 119889119909+ 1199011205832 int

Ω11990611990121198911minus1199012119901 1198922 (1199061 1199062) 119889119909

(41)

where 119891119894119901(V) is defined by

119891119894119901 (V) = exp( 11 minus 120591 ((119901 minus 1) 119901) intV

V120594119894 (119904) 119889119904)

119894 = 1 2(42)

Proof It is easy to check

11989110158401119901 (V) = 11 minus 120591 ((119901 minus 1) 119901)1205941 (V) 1198911119901 (V) 11989110158402119901 (V) = 11 minus 120591 ((119901 minus 1) 119901)1205942 (V) 1198912119901 (V)

(43)

Then for any 119901 gt 1 we have119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909 = 119901int

Ω119906119901minus11 11990611199051198911minus1199011119901 119889119909 + (1 minus 119901)

sdot intΩ1199061199011119891minus1199011119901 11989110158401119901V119905119889119909 = 119901int

Ω119906119901minus11 1198911minus1199011119901 [Δ1199061 minus nabla

sdot (11990611205941 (V) nablaV) + 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1)11990611205941 (V) ΔV]119889119909

+ 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990111198911minus1199011119901 1205941 (V)

sdot ℎ (1199061 1199062 V) 119889119909+ 1199011205831 int

Ω11990611990111198911minus1199011119901 1198921 (1199061 1199062) 119889119909 fl 1198681 + 1198682 + 1198683

(44)

Clearly if 1198681 le 0 then (40) is a consequence of (44) We inthe following verify that the result holds

In fact

1198681 = 119901intΩ119906119901minus11 1198911minus1199011119901 [Δ1199061 minus nabla sdot (11990611205941 (V) nablaV)

+ 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1) (nabla sdot (11990611205941 (V) nablaV) minus nabla (11990611205941 (V))

sdot nablaV)] 119889119909 = 119901intΩ119906119901minus11 1198911minus1199011119901 [nabla

sdot (1198911119901nabla (1199061119891minus11119901 )) minus 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1) (1205941 (V) nabla1199061 sdot nablaV

+ 119906112059410158401 (V) |nablaV|2)] 119889119909 = 119901 (1 minus 119901)

sdot intΩ119906119901minus21 1198912minus1199011119901 [1198911119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162

minus 1205911199061 (1205941 (V) nabla1199061 sdot nablaV + 119906112059410158401 (V) |nablaV|2)1198911119901 (119901 minus 120591 (119901 minus 1)) ] 119889119909= 119901 (1 minus 119901)int

Ω119906119901minus21 1198913minus1199011119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162 119889119909

+ 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ 119906119901minus11 1198911minus1199011119901 1205941 (V) nabla1199061 sdot nablaV 119889119909

+ 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ 11990611990111198911minus1199011119901 12059410158401 (V) |nablaV|2 119889119909 fl 1198691

+ 1198692 + 1198693

(45)

Since nabla1199061 = nabla(1198911119901(1199061119891minus11119901 )) multiplying both sides by1205941(V)nablaV we get1205941 (V) nabla1199061 sdot nablaV

= 11989111199011205941 (V) nabla (1199061119891minus11119901 ) sdot nablaV+ 11 minus 120591 ((119901 minus 1) 119901)119906112059421 (V) |nablaV|2

(46)

and by 120576-Youngrsquos inequality1205941 (V) nabla (1199061119891minus11119901 ) sdot nablaV

le 119901 minus 120591 (119901 minus 1)120591

1198911119901119906110038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162

+ 120591119901 minus 120591 (119901 minus 1) 1199061119891111990112059421 (V) |nablaV|2

(47)

Combining (46) and (47) and substituting it into 1198692 we derive1198692 le 119901 (119901 minus 1)int

Ω119906119901minus21 1198913minus1199011119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162 119889119909

+ 120591119901 (119901 minus 1) (120591 + 119901)(119901 minus 120591 (119901 minus 1))2 int

Ω11990611990111198911minus1199011119901 12059421 (V) |nablaV|2 119889119909

(48)

Discrete Dynamics in Nature and Society 7

Inserting (48) into (45) yields

1198681 le 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ (

120591 + 119901(119901 minus 120591 (119901 minus 1))12059421 (V)

+ 12059410158401 (V)) 11990611990111198911minus1199011119901 |nablaV|2 119889119909 = 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1)

sdot intΩ[ 1(1 minus 120591) (1 minus 1205912120591 + (1 minus 120591) 119901)12059421 (V)

+ 12059410158401 (V)] 11990611990111198911minus1199011119901 |nablaV|2 119889119909 le 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1)

sdot intΩ( 1(1 minus 120591)12059421 (V) + 12059410158401 (V)) 11990611990111198911minus1199011119901 |nablaV|2 119889119909

(49)

we deduce from assumption (11) that 1198681 le 0 for any 119901 gt 1and the proof ends In the same way we achieve (41) Thiscompletes the proof of Lemma 6

In view of V isin 119862((0 119879max) 1198621(Ω)) and initial date V ltV0 lt V we consider 119879lowast gt 0 defined by

119879lowast fl 119879max if V le V in (0 119879max) 119879V if V119871infin(Ω119879max )

gt V (50)

where 119879V lt 119879max satisfies that V119871infin(Ω119879V ) le V andV119871infin(Ω119879V+120575) gt V for any 120575 gt 0Lemma7 Letting119901 gt 1 and for any 119905 lt 119879lowast then the solutionsto (1) satisfy

1119901 minus 1 119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909le minus (12058101 + 1205831) int

Ω119906119901+11 119891minus1199011119901 119889119909

+ ( 12058111 minus 120591 + 119901119901 minus 11205831)intΩ11990611990111198911minus1199011119901 119889119909

(51)

1119901 minus 1 119889119889119905 intΩ 11990611990121198911minus1199012119901 119889119909le minus (12058102 + 1205832) int

Ω119906119901+12 119891minus1199012119901 119889119909

+ ( 12058121 minus 120591 + 119901119901 minus 11205832)intΩ11990611990121198911minus1199012119901 119889119909

(52)

Proof In order to prove Lemma 7 we consider the terms 1198682and 1198683 appearing in the proof of Lemma 6

By the nonnegativity for 1199061 1199062 and assumption (6) weobtain

minusℎ (1199061 1199062 V) le minus1199061 + 12058111205941 (V) (53)

Multiplying (53) by 11990611990111198911minus1199011119901 1205941(V) and integrating overΩ yield

minus intΩ11990611990111198911minus1199011119901 1205941 (V) ℎ (1199061 1199062 V) 119889119909

le minusintΩ119906119901+11 1198911minus1199011119901 1205941 (V) 119889119909 + 1205811 int

Ω11990611990111198911minus1199011119901 119889119909

(54)

By the monotonic property of exponential function andassumption (7) we have

0 lt 12058101 le 1205941 (V) exp(intV

V1205941 (119904) 119889119904) le 1205941 (V) 1198911119901 (V) (55)

It follows that (54) becomes

minus intΩ11990611990111198911minus1199011119901 1205941 (V) ℎ (1199061 1199062 V) 119889119909

le minus12058101 intΩ119906119901+11 119891minus1199011119901 119889119909 + 1205811 int

Ω11990611990111198911minus1199011119901 119889119909

(56)

Multiplying (56) by (119901 minus 1)(1 minus 120591((119901 minus 1)119901)) we get1198682 le 119901 minus 1

1 minus 120591 ((119901 minus 1) 119901) (minus12058101 intΩ 119906119901+11 119891minus1199011119901 119889119909+ 1205811 int

Ω11990611990111198911minus1199011119901 119889119909) le (119901 minus 1)

sdot (minus12058101 intΩ119906119901+11 119891minus1199011119901 119889119909 + 12058111 minus 120591 int

Ω11990611990111198911minus1199011119901 119889119909)

(57)

For the term 1198683 we easily achieve1198683 le 1199011205831 (int

Ω11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

le (119901 minus 1) 1205831 (intΩ11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

+ 1205831 intΩ11990611990111198911minus1199011119901 119889119909

(58)

Combining (44) (57) and (58) leads to

119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909 le (119901 minus 1)sdot (minus12058101 int

Ω119906119901+11 119891minus1199011119901 119889119909 + 12058111 minus 120591 int

Ω11990611990111198911minus1199011119901 119889119909)

+ (119901 minus 1) 1205831 (intΩ11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

+ 1205831 intΩ11990611990111198911minus1199011119901 119889119909

(59)

which implies that (51) holds The same arguments are usedto achieve (52) The proof ends

To establish the uniform boundedness of solutions weapply an iterative method based on the 119871119901-norm of solutionsinspired by the Moser-Alikakos type iterations (see [10 13])

8 Discrete Dynamics in Nature and Society

Lemma 8 For any 119905 lt 119879lowast then the solutions to (1) arebounded by

100381710038171003817100381711990611003817100381710038171003817119871infin(Ω) le 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

(60)

100381710038171003817100381711990621003817100381710038171003817119871infin(Ω) le 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(61)

Proof DenoteintΩ 11990611990111198911minus1199011119901 119889119909 byΨ(119905) Taking arbitrary positiveconstant 120588 gt 0 then it holds that

Ψ (119905) = int1198911119901le120588(12058101+1205831)1199061

11990611990111198911minus1199011119901 119889119909+ int1198911119901gt120588(12058101+1205831)1199061

11990611990111198911minus1199011119901 119889119909le 120588 (12058101 + 1205831) int

1198911119901le120588(12058101+1205831)1199061119906119901+11 119891minus1199011119901 119889119909

+ (120588 (12058101 + 1205831))1minus119901 int1198911119901gt120588(12058101+1205831)1199061

1199061119889119909le 120588 (12058101 + 1205831) int

Ω119906119901+11 119891minus1199011119901 119889119909

+ (120588 (12058101 + 1205831))1minus119901 intΩ1199061119889119909

(62)

Hence

minus (12058101 + 1205831) intΩ119906119901+11 119891minus1199011119901 119889119909

le minus1120588Ψ (119905) + 120588minus119901 (12058101 + 1205831)1minus119901 intΩ1199061119889119909

(63)

Substituting (63) into (51) we have

1119901 minus 1 119889119889119909Ψ (119905) le ( 12058111 minus 120591 + 119901119901 minus 11205831 minus 1120588)Ψ (119905)+ 120588minus119901 (12058101 + 1205831)1minus119901 int

Ω1199061119889119909

(64)

Take 1120588 gt 1205811(1 minus 120591) + (119901(119901 minus 1))1205831 Noticing Lemma 5and initial date 11990610 isin 1198621(Ω) we apply the Gronwall Lemmato obtain

Ψ (119905) le max120588minus119901 (12058101 + 1205831)1minus119901

sdot ( 1120588 minus 12058111 minus 120591 minus 119901119901 minus 11205831)minus1 intΩ1199061119889119909 Ψ (0)

(65)

taking limits leads to

lim119901rarrinfin

Ψ1119901 (119905) le max 120588minus1 (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω) (66)

Since

Ψ (119905) = intΩ11990611990111198911minus1199011119901 (V) 119889119909 ge 1198911minus1199011119901 (V) int

Ω1199061199011119889119909 (67)

combining (66) and (67) and taking limits as 120588minus1 rarr 1205811(1 minus120591) + 1205831 it is found that (60) holds The same processes areused to prove (61) which ends the proof of Lemma 8

Remark 9 In the above arguments we used well such factLetΩ be bounded and regular domain ofR119899 If 119906119871119901(Ω) le 119862where119862 is some positive constant and independent of119901 then119906119871119901(Ω) rarr 119906119871infin(Ω) as 119901 rarr infin

To end the arguments of uniform boundedness of solu-tions it follows to prove that 119879lowast = 119879max and 119879max = +infin bycontradiction

As in Lemmas 4ndash8 we define 1 2 and V for the uniquesolutions in (0 max) to

1119905 = Δ1 minus nabla sdot (11205941 (V) nablaV) + 120583111198921 (1 2) 119909 isin Ω 119905 gt 0

2119905 = Δ2 minus nabla sdot (21205942 (V) nablaV) + 120583211990621198922 (1 2) 119909 isin Ω 119905 gt 0

V119905 = 120591ΔV + ℎ (1 2 V) 119909 isin Ω 119905 gt 01205971120597] = 1205972120597] = 120597V120597] = 0 119909 isin 120597Ω 119905 gt 0

1 (119909 0) = 11990610 (119909) 2 (119909 0) = 11990620 (119909) V (119909 0) = V0 (119909)

119909 isin Ω

(68)

where max is defined in the same fashion as (32) and thetruncation ℎ(1199061 1199062 V) is defined as follows

ℎ (1199061 1199062 V) fl ℎ (1199061 1199062 V) if V le Vℎ (1199061 1199062 V) if V gt V (69)

Then it is easy to see that100381710038171003817100381711003817100381710038171003817119871infin(Ω) le 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

100381710038171003817100381721003817100381710038171003817119871infin(Ω) le 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(70)

as far as V le VBy analogy with 119879lowast we define

lowast fl max if V le V in (0 max) V if V119871infin(Ωmax )

gt V (71)

Discrete Dynamics in Nature and Society 9

where V lt max satisfies that V119871infin(ΩV ) le V and

V119871infin(ΩV+120575) gt V for any 120575 gt 0 To prove lowast = max wesuppose

lowast lt max (72)

then ℎ(1 2 V) le ℎ(1199061 1199062 V) = ℎ(1199061 1199062 V) lt 0 It follows toeasily check that V is a strict upper-solution to

V119905 minus 120591ΔV = ℎ (1 2 V) in Ω times (0 lowast) 120597V120597] = 0 on 120597Ω times (0 lowast)

V (119909 0) = V0 (119909) in Ω(73)

This implies V(119909 119905) lt V forall119909 isin Ω 119905 isin (0 lowast) whichcontradicts (72) and proves lowast = max as well as max = infin

Thanks to V le V we show that (1 2 V) is also a solutionto (6) So we have

119879lowast = 119879max ge max = infin (74)

The proof of Theorem 3 is completed

Competing Interests

The author declares that they have no competing interests

Acknowledgments

Thiswork is supported by the ChinaNational Natural ScienceFoundation (nos 11361055 and 11261053) and the NaturalScience ResearchKey Project of University of Anhui Province(no KJ2015A251)

References

[1] J D Murray Mathematical Biology vol 19 Springer BerlinGermany 2nd edition 1993

[2] M Winkler ldquoBoundedness in the higher-dimensionalparabolic-parabolic chemotaxis system with logistic sourcerdquoCommunications in Partial Differential Equations vol 35 no 8pp 1516ndash1537 2010

[3] T Hillen and K J Painter ldquoA userrsquos guide to PDE models forchemotaxisrdquo Journal of Mathematical Biology vol 58 no 1-2pp 183ndash217 2009

[4] E F Keller and L A Segel ldquoInitiation of slimemold aggregationviewed as an instabilityrdquo Journal of Theoretical Biology vol 26no 3 pp 399ndash415 1970

[5] M Winkler ldquoAbsence of collapse in a parabolic chemo-taxis system with signal-dependent sensitivityrdquo MathematischeNachrichten vol 283 no 11 pp 1664ndash1673 2010

[6] J I Tello and M Winkler ldquoStabilization in a two-specieschemotaxis system with a logistic sourcerdquo Nonlinearity vol 25no 5 pp 1413ndash1425 2012

[7] M Negreanu and J I Tello ldquoAsymptotic stability of a twospecies chemotaxis systemwith non-diffusive chemoattractantrdquoJournal of Differential Equations vol 258 no 5 pp 1592ndash16172015

[8] Q Zhang and Y Li ldquoGlobal boundedness of solutions toa two-species chemotaxis systemrdquo Zeitschrift fur angewandteMathematik und Physik vol 66 no 1 pp 83ndash93 2015

[9] C Mu L Wang P Zheng and Q Zhang ldquoGlobal existenceand boundedness of classical solutions to a parabolic-parabolicchemotaxis systemrdquo Nonlinear Analysis Real World Applica-tions vol 14 no 3 pp 1634ndash1642 2013

[10] Y Tao ldquoBoundedness in a chemotaxis model with oxygenconsumption by bacteriardquo Journal ofMathematical Analysis andApplications vol 381 no 2 pp 521ndash529 2011

[11] D Horstmann and M Winkler ldquoBoundedness vs blow-up in achemotaxis systemrdquo Journal of Differential Equations vol 215no 1 pp 52ndash107 2005

[12] M Winkler ldquoAggregation vs global diffusive behavior in thehigher-dimensional Keller-Segel modelrdquo Journal of DifferentialEquations vol 248 no 12 pp 2889ndash2905 2010

[13] PQuittner andP Souplet Superlinear Parabolic Problems Blow-Up Global Existence and Steady States Birkhauser AdvancedTexts Birkhauser Basel Switzerland 2007

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Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

(iv) Assume that

ℎ (1199061 1199062 V) ge 0ℎ (1199061 1199062 V) lt 0

for 0 le 1199061 le 1199061 0 le 1199062 le 1199062(8)

where

1199061 fl 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

1199062 fl 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(9)

for 119891119894(V) defined by

119891119894 (V) = exp( 11 minus 120591 intV

V120594119894 (119904) 119889119904) 119894 = 1 2 (10)

(v) Finally for some technical reasons we also assumethat

1205942119894 (V) + (1 minus 120591) 1205941015840119894 (V) le 0 119894 = 1 2 (11)

We illustrate the validity to above assumptions with thefollowing generalized example [7]

Example 1 We take the chemosensitivity functions 120594119894(V) =120572119894(120573119894 + V) (119894 = 1 2) for positive constants 120572119894 120573119894 fulfilling120572119894 le 1 minus 1205914120574 lt 120573119894 le 8120574

119894 = 1 2(12)

Clearly (11) holds Take a lower bound V = 0 and upper boundV is to be defined later Moreover the initial dates 11990610 and 11990620are satisfied as

1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω) le ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω) le ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1

(13)

Then consider the following

(1) Condition (6) is equivalent to 120574120572119894(V(120573119894 + V)) le 120581119894 sochoosing 120581119894 = 120574120572119894(V(120573119894 + V)) (119894 = 1 2)

(2) Taking positive constants 1205810119894 = 120572119894120573minus120572119894119894 (120573119894+V)1minus120572119894 (119894 =1 2) then condition (7) holds since

1205810119894 le 120572119894120573119894 + V119890120572119894 intV0 (1(120573119894+119904))119889119904 = 120572119894(120573119894 + V)1minus120572119894 (

1120573119894)120572119894

(14)

for 119894 = 1 2

(3) We notice that ℎ(0 0 0) = 0 as well asℎ (1199061 1199062 V) = 1199061 + 1199062 minus 120574V

= 119890(1205721(1minus120591)) intV0 (1(1205731+119904))119889119904 1205811 (1 minus 120591) + 120583112058101 + 1205831+ 119890(1205722(1minus120591)) intV0 (1(1205732+119904))119889119904 1205812 (1 minus 120591) + 120583212058102 + 1205832 minus 120574V

(15)

A sufficient condition for the second inequality in (8)holding is

119890(1205721(1minus120591)) intV0 (1(1205731+119904))119889119904 1205811 (1 minus 120591) + 120583112058101 + 1205831 lt 1205742V119890(1205722(1minus120591)) intV0 (1(1205732+119904))119889119904 1205812 (1 minus 120591) + 120583212058102 + 1205832 lt 1205742V

(16)

and for simplicity let us take 120583119894 = 120581119894(1minus120591) (119894 = 1 2)and then derive

V gt max 1205731(1205744) 1205731 minus 1 1205732(1205744) 1205732 minus 1 (17)

Up to now the above all restrictive conditions are verifiedwhich implies that conditions (6)ndash(11) are sufficient to ensurethe global existence of solutions

Remark 2 In literature [7] the chemical production functionℎ(1199061 1199062 V) = minusV + 1199061 + 1199062 and the chemotactic sensitivity120594119894(V) = 120574119894(1 + 120574119894V) (119894 = 1 2)In this paper the purpose is to study the global existence

and uniform boundedness of solutions to (1) applying aniterative methodThemain results are stated by the followingtheorem

Theorem 3 Under assumptions (6)ndash(11) for any initial date(11990610 11990620 V0) isin (1198621(Ω))3 satisfying the homogeneous Neu-mann boundary conditions and 11990610 ge 0 11990620 ge 0 there exists aunique solution to (1)

(1199061 1199062 V)isin (119871119901 ([0infin) 1198822119901 (Ω)) cap1198821119901 ([0infin) 119871119901 (Ω)))3 (18)

for any 119901 gt 119899 Moreover the solution is uniformly boundedthat is

100381710038171003817100381711990611003817100381710038171003817119871infin(Ω) + 100381710038171003817100381711990621003817100381710038171003817119871infin(Ω) + V119871infin(Ω) le 119862 (19)

where 119862 (ltinfin) is positive constantThe proof of Theorem 3 is split into several steps Step

one we start to consider the local existence of solutions andthen proceed with the extensibility criterion Step two undersome appropriate conditions we prove that the solutions areuniformly bounded in time

4 Discrete Dynamics in Nature and Society

3 Global Existence of Solutions

We will first be devoted to dealing with local-in-time exis-tence and uniqueness of a nonnegative solution for (1) Thecorresponding conclusions are written by Lemma 4

31 Local Existence of Solutions

Lemma 4 Given positive initial date (11990610 11990620 V0) isin (1198621(Ω))3and the same assumptions as inTheorem 3 there exists a smallenough time 119879 gt 0 and a unique triple of nonnegativefunctions (1199061 1199062 V) isin (119871119901([0 119879]1198822119901(Ω)) cap 1198821119901([0 119879]119871119901(Ω)))3 (for 119901 gt 119899) such that (1199061 1199062 V) is a solution of (1)in Ω times (0 119879) Moreover we have V ge V

Proof Introduce the change of variables 1 and 2 given by

1199061 = 1198911 (V) 11199062 = 1198912 (V) 2 (20)

where the function 119891119894(V) is defined by

119891119894 (V) = exp 11 minus 120591 intV

V120594119894 (119904) 119889119904 119894 = 1 2 (21)

A direct calculation yields

1199061119905 = 11989110158401 (V) (120591ΔV + ℎ (1199061 1199062 V)) 1+ 1198911 (V) 1119905

Δ1199061 = 119891101584010158401 (V) 1 |nablaV|2 + 211989110158401 (V) nabla1 sdot nablaV+ 11989110158401 (V) 1ΔV + 1198911 (V) Δ1

nabla sdot (11990611205941 (V) nablaV) = 11989110158401 (V) 11205941 (V) |nablaV|2+ 1198911 (V) 1205941 (V) nabla1 sdot nablaV+ 1198911 (V) 112059410158401 (V) |nablaV|2+ 1198911 (V) 11205941 (V) ΔV

(22)

similarly

1199062119905 = 11989110158402 (V) (120591ΔV + ℎ (1199061 1199062 V)) 2+ 1198912 (V) 2119905

Δ1199062 = 119891101584010158402 (V) 2 |nablaV|2 + 211989110158402 (V) nabla2 sdot nablaV+ 11989110158402 (V) 2ΔV + 1198912 (V) Δ2

nabla sdot (11990621205942 (V) nablaV) = 11989110158402 (V) 21205942 (V) |nablaV|2+ 1198912 (V) 1205942 (V) nabla2 sdot nablaV+ 1198912 (V) 212059410158402 (V) |nablaV|2+ 1198912 (V) 21205942 (V) ΔV

(23)

where

1198911015840119894 (V) = 120594119894 (V)1 minus 120591119891119894 (V) 11989110158401015840119894 (V) = ( 1205942119894 (V)(1 minus 120591)2 +

1205941015840119894 (V)1 minus 120591 )119891119894 (V) 119894 = 1 2

(24)

Substitute (20) (22) and (23) into (1) and denote that

L119894 (V) 119894 = 119894119905 minus Δ119894 minus 1 + 1205911 minus 120591120594119894 (V) nablaV sdot nabla119894119866119894 (1 2 V) = minus119909119894 (V)1 minus 120591 ℎ (1198911 (V) 1 1198912 (V) 2 V)+ 120583119894119892119894 (1198911 (V) 1 1198912 (V) 2)

(25)

for 119894 = 1 2Then system (1) becomes

L1 (V) 1= 11198661 (1 2 V)

+ 1 120591(1 minus 120591)2 (12059421 (V) + (1 minus 120591) 12059410158401 (V)) |nablaV|2

119909 isin Ω 119905 gt 0L2 (V) 2

= 21198662 (1 2 V)+ 2 120591

(1 minus 120591)2 (12059422 (V) + (1 minus 120591) 12059410158402 (V)) |nablaV|2 119909 isin Ω 119905 gt 0

V119905 = 120591ΔV + ℎ (1198911 (V) 1 1198912 (V) 2 V) 119909 isin Ω 119905 gt 0

(26)

with boundary conditions 1205971120597] = 1205972120597] = 120597V120597] = 0119909 isin 120597Ω 119905 gt 0 and initial conditions 1(119909 0) fl 10(119909) =11990610(119909)1198911(V0(119909)) 2(119909 0) fl 20(119909) = 11990620(119909)1198912(V0(119909))V(119909 0) = V0(119909) 119909 isin Ω

For sufficiently small 119879 gt 0 in the space [119871119901([0 119879]1198822119901(Ω)) cap 1198821119901([0 119879] 119871119901(Ω))]2 (for 119901 gt 119899) we implementa fixed point argument Taking V isin 119862([0 119879] 1198621(Ω)) satisfyingV isin (V V) and |nablaV| lt 119862 we define 1 2 in (0 119879) as the uniquesolution to the questionL1 (V) 1

= 11198661 (1 2 V)+ 1 120591

(1 minus 120591)2 (12059421 (V) + (1 minus 120591) 12059410158401 (V)) |nablaV|2in Ω times (0 119879)

L2 (V) 2= 21198662 (1 2 V)

+ 2 120591(1 minus 120591)2 (12059422 (V) + (1 minus 120591) 12059410158402 (V)) |nablaV|2

in Ω times (0 119879)

(27)

Discrete Dynamics in Nature and Society 5

for the details we refer the reader to [13 Remark 483]Noticethat the right-hand side terms of (27) are the multiplicativefor 1 and 2 applying the maximum principle to verify thatboth 1 and 2 are nonnegative

Since condition (11) we easily see that the regular func-tions 1 and 2 satisfy

L1 (V) 1 le 11198661 (1 2 V) L2 (V) 2 le 21198662 (1 2 V) (28)

and the right-hand side terms of (27) are quasi-decreasedlooking upon the lower-solution 1199061 = 1199062 = 0 one mayconstruct upper-solutions 1199061 and 1199062 such that

0 le 1199061 le 11990610 le 1199062 le 1199062119905 lt 119879

(29)

It follows that we apply Theorem 21 of [7] to get a uniquenonnegative solution

We solve the parabolic equation

V119905 = 120591ΔV + ℎ (1198911 (V) 1 1198912 (V) 2 V) (30)

Thanks to the essential estimations of parabolic equations andthe embedding theorems we apply the Schauder fixed pointtheorem to obtain the local existence of solution V(119909 119905) Thesmoothness of ℎ(1198911(V)1 1198912(V)2 V) ensures the uniquenessof solution V(119909 119905)

To prove V ge V in view of assumption (8) and themonotone of ℎ(1199061 1199062 V) we have 0 le ℎ(0 0 V) le ℎ(1199061 1199062 V)It follows that V is a lower-solution for the equation

V119905 minus 120591ΔV = ℎ (1199061 1199062 V) in Ω times (0 119879) 120597V120597] = 0 on 120597Ω times (0 119879)

V (119909 0) = V0 (119909) in Ω(31)

The proof of Lemma 4 is completed

About the above solution (1199061 1199062 V) we have the followingextensibility criterionThe solution is extended to the interval(0 119879max) where 119879max has the following property

If 119879max lt infinthen lim sup

119905rarr119879max

(10038171003817100381710038171199061 (sdot 119905)1003817100381710038171003817119871infin(Ω) + 10038171003817100381710038171199062 (sdot 119905)1003817100381710038171003817119871infin(Ω)+ V (sdot 119905)119871infin(Ω)) = infin

(32)

32 Uniform Boundedness of Solutions To obtain some a pri-ori estimates we need some technical lemmasThe following1198711-estimate of solution is first given

Lemma 5 For all 119905 isin (0 119879119898119886119909) the solutions to (1) satisfy thefollowing estimates

1003817100381710038171003817119906110038171003817100381710038171198711(Ω) le max |Ω| 10038171003817100381710038171199061010038171003817100381710038171198711(Ω) le 1198721003817100381710038171003817119906210038171003817100381710038171198711(Ω) le max |Ω| 10038171003817100381710038171199062010038171003817100381710038171198711(Ω) le 119872V1198711(Ω) le max 1003817100381710038171003817V010038171003817100381710038171198711(Ω) 2119872120574minus1

(33)

where119872 = max|Ω| 119906101198711(Ω) 119906201198711(Ω)Proof Integrating the first equation of (1) overΩ we have

119889119889119905 intΩ 1199061119889119909 = 1205831 intΩ(1199061 minus 11990621 minus 119886111990611199062) 119889119909

le 1205831 intΩ(1199061 minus 11990621) 119889119909

(34)

Using Cauchy inequality yields

119889119889119905 intΩ 1199061119889119909 le 1205831 intΩ(1 minus 1199061) 119889119909

le 1205831 |Ω| minus 1205831 intΩ1199061119889119909

(35)

By the Gronwall Lemma we derive

10038171003817100381710038171199061 (119909 119905)10038171003817100381710038171198711(Ω) le max |Ω| 100381710038171003817100381711990610 (119909)10038171003817100381710038171198711(Ω) le 119872 (36)

a similar estimate holds for 1199062 which leads to

10038171003817100381710038171199062 (119909 119905)10038171003817100381710038171198711(Ω) le max |Ω| 100381710038171003817100381711990620 (119909)10038171003817100381710038171198711(Ω) le 119872 (37)

Integrating the third equation of (1) overΩ we have

119889119889119905 intΩ V 119889119909 = intΩ(minus120574V + 1199061 + 1199062) 119889119909

= minus120574intΩV 119889119909 + int

Ω(1199061 + 1199062) 119889119909

le minus120574intΩV 119889119909 + 2119872

(38)

it implies

V1198711(Ω) le max 1003817100381710038171003817V010038171003817100381710038171198711(Ω) 2119872120574minus1 (39)

This proves the lemma

6 Discrete Dynamics in Nature and Society

Lemma 6 Letting 119901 gt 1 and under assumption (11) then thefollowing estimates hold

119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909le 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990111205941 (V) 1198911minus1199011119901 ℎ (1199061 1199062 V) 119889119909+ 1199011205831 int

Ω11990611990111198911minus1199011119901 1198921 (1199061 1199062) 119889119909

(40)

119889119889119905 intΩ 11990611990121198911minus1199012119901 119889119909le 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990121205942 (V) 1198911minus1199012119901 ℎ (1199061 1199062 V) 119889119909+ 1199011205832 int

Ω11990611990121198911minus1199012119901 1198922 (1199061 1199062) 119889119909

(41)

where 119891119894119901(V) is defined by

119891119894119901 (V) = exp( 11 minus 120591 ((119901 minus 1) 119901) intV

V120594119894 (119904) 119889119904)

119894 = 1 2(42)

Proof It is easy to check

11989110158401119901 (V) = 11 minus 120591 ((119901 minus 1) 119901)1205941 (V) 1198911119901 (V) 11989110158402119901 (V) = 11 minus 120591 ((119901 minus 1) 119901)1205942 (V) 1198912119901 (V)

(43)

Then for any 119901 gt 1 we have119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909 = 119901int

Ω119906119901minus11 11990611199051198911minus1199011119901 119889119909 + (1 minus 119901)

sdot intΩ1199061199011119891minus1199011119901 11989110158401119901V119905119889119909 = 119901int

Ω119906119901minus11 1198911minus1199011119901 [Δ1199061 minus nabla

sdot (11990611205941 (V) nablaV) + 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1)11990611205941 (V) ΔV]119889119909

+ 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990111198911minus1199011119901 1205941 (V)

sdot ℎ (1199061 1199062 V) 119889119909+ 1199011205831 int

Ω11990611990111198911minus1199011119901 1198921 (1199061 1199062) 119889119909 fl 1198681 + 1198682 + 1198683

(44)

Clearly if 1198681 le 0 then (40) is a consequence of (44) We inthe following verify that the result holds

In fact

1198681 = 119901intΩ119906119901minus11 1198911minus1199011119901 [Δ1199061 minus nabla sdot (11990611205941 (V) nablaV)

+ 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1) (nabla sdot (11990611205941 (V) nablaV) minus nabla (11990611205941 (V))

sdot nablaV)] 119889119909 = 119901intΩ119906119901minus11 1198911minus1199011119901 [nabla

sdot (1198911119901nabla (1199061119891minus11119901 )) minus 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1) (1205941 (V) nabla1199061 sdot nablaV

+ 119906112059410158401 (V) |nablaV|2)] 119889119909 = 119901 (1 minus 119901)

sdot intΩ119906119901minus21 1198912minus1199011119901 [1198911119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162

minus 1205911199061 (1205941 (V) nabla1199061 sdot nablaV + 119906112059410158401 (V) |nablaV|2)1198911119901 (119901 minus 120591 (119901 minus 1)) ] 119889119909= 119901 (1 minus 119901)int

Ω119906119901minus21 1198913minus1199011119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162 119889119909

+ 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ 119906119901minus11 1198911minus1199011119901 1205941 (V) nabla1199061 sdot nablaV 119889119909

+ 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ 11990611990111198911minus1199011119901 12059410158401 (V) |nablaV|2 119889119909 fl 1198691

+ 1198692 + 1198693

(45)

Since nabla1199061 = nabla(1198911119901(1199061119891minus11119901 )) multiplying both sides by1205941(V)nablaV we get1205941 (V) nabla1199061 sdot nablaV

= 11989111199011205941 (V) nabla (1199061119891minus11119901 ) sdot nablaV+ 11 minus 120591 ((119901 minus 1) 119901)119906112059421 (V) |nablaV|2

(46)

and by 120576-Youngrsquos inequality1205941 (V) nabla (1199061119891minus11119901 ) sdot nablaV

le 119901 minus 120591 (119901 minus 1)120591

1198911119901119906110038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162

+ 120591119901 minus 120591 (119901 minus 1) 1199061119891111990112059421 (V) |nablaV|2

(47)

Combining (46) and (47) and substituting it into 1198692 we derive1198692 le 119901 (119901 minus 1)int

Ω119906119901minus21 1198913minus1199011119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162 119889119909

+ 120591119901 (119901 minus 1) (120591 + 119901)(119901 minus 120591 (119901 minus 1))2 int

Ω11990611990111198911minus1199011119901 12059421 (V) |nablaV|2 119889119909

(48)

Discrete Dynamics in Nature and Society 7

Inserting (48) into (45) yields

1198681 le 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ (

120591 + 119901(119901 minus 120591 (119901 minus 1))12059421 (V)

+ 12059410158401 (V)) 11990611990111198911minus1199011119901 |nablaV|2 119889119909 = 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1)

sdot intΩ[ 1(1 minus 120591) (1 minus 1205912120591 + (1 minus 120591) 119901)12059421 (V)

+ 12059410158401 (V)] 11990611990111198911minus1199011119901 |nablaV|2 119889119909 le 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1)

sdot intΩ( 1(1 minus 120591)12059421 (V) + 12059410158401 (V)) 11990611990111198911minus1199011119901 |nablaV|2 119889119909

(49)

we deduce from assumption (11) that 1198681 le 0 for any 119901 gt 1and the proof ends In the same way we achieve (41) Thiscompletes the proof of Lemma 6

In view of V isin 119862((0 119879max) 1198621(Ω)) and initial date V ltV0 lt V we consider 119879lowast gt 0 defined by

119879lowast fl 119879max if V le V in (0 119879max) 119879V if V119871infin(Ω119879max )

gt V (50)

where 119879V lt 119879max satisfies that V119871infin(Ω119879V ) le V andV119871infin(Ω119879V+120575) gt V for any 120575 gt 0Lemma7 Letting119901 gt 1 and for any 119905 lt 119879lowast then the solutionsto (1) satisfy

1119901 minus 1 119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909le minus (12058101 + 1205831) int

Ω119906119901+11 119891minus1199011119901 119889119909

+ ( 12058111 minus 120591 + 119901119901 minus 11205831)intΩ11990611990111198911minus1199011119901 119889119909

(51)

1119901 minus 1 119889119889119905 intΩ 11990611990121198911minus1199012119901 119889119909le minus (12058102 + 1205832) int

Ω119906119901+12 119891minus1199012119901 119889119909

+ ( 12058121 minus 120591 + 119901119901 minus 11205832)intΩ11990611990121198911minus1199012119901 119889119909

(52)

Proof In order to prove Lemma 7 we consider the terms 1198682and 1198683 appearing in the proof of Lemma 6

By the nonnegativity for 1199061 1199062 and assumption (6) weobtain

minusℎ (1199061 1199062 V) le minus1199061 + 12058111205941 (V) (53)

Multiplying (53) by 11990611990111198911minus1199011119901 1205941(V) and integrating overΩ yield

minus intΩ11990611990111198911minus1199011119901 1205941 (V) ℎ (1199061 1199062 V) 119889119909

le minusintΩ119906119901+11 1198911minus1199011119901 1205941 (V) 119889119909 + 1205811 int

Ω11990611990111198911minus1199011119901 119889119909

(54)

By the monotonic property of exponential function andassumption (7) we have

0 lt 12058101 le 1205941 (V) exp(intV

V1205941 (119904) 119889119904) le 1205941 (V) 1198911119901 (V) (55)

It follows that (54) becomes

minus intΩ11990611990111198911minus1199011119901 1205941 (V) ℎ (1199061 1199062 V) 119889119909

le minus12058101 intΩ119906119901+11 119891minus1199011119901 119889119909 + 1205811 int

Ω11990611990111198911minus1199011119901 119889119909

(56)

Multiplying (56) by (119901 minus 1)(1 minus 120591((119901 minus 1)119901)) we get1198682 le 119901 minus 1

1 minus 120591 ((119901 minus 1) 119901) (minus12058101 intΩ 119906119901+11 119891minus1199011119901 119889119909+ 1205811 int

Ω11990611990111198911minus1199011119901 119889119909) le (119901 minus 1)

sdot (minus12058101 intΩ119906119901+11 119891minus1199011119901 119889119909 + 12058111 minus 120591 int

Ω11990611990111198911minus1199011119901 119889119909)

(57)

For the term 1198683 we easily achieve1198683 le 1199011205831 (int

Ω11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

le (119901 minus 1) 1205831 (intΩ11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

+ 1205831 intΩ11990611990111198911minus1199011119901 119889119909

(58)

Combining (44) (57) and (58) leads to

119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909 le (119901 minus 1)sdot (minus12058101 int

Ω119906119901+11 119891minus1199011119901 119889119909 + 12058111 minus 120591 int

Ω11990611990111198911minus1199011119901 119889119909)

+ (119901 minus 1) 1205831 (intΩ11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

+ 1205831 intΩ11990611990111198911minus1199011119901 119889119909

(59)

which implies that (51) holds The same arguments are usedto achieve (52) The proof ends

To establish the uniform boundedness of solutions weapply an iterative method based on the 119871119901-norm of solutionsinspired by the Moser-Alikakos type iterations (see [10 13])

8 Discrete Dynamics in Nature and Society

Lemma 8 For any 119905 lt 119879lowast then the solutions to (1) arebounded by

100381710038171003817100381711990611003817100381710038171003817119871infin(Ω) le 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

(60)

100381710038171003817100381711990621003817100381710038171003817119871infin(Ω) le 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(61)

Proof DenoteintΩ 11990611990111198911minus1199011119901 119889119909 byΨ(119905) Taking arbitrary positiveconstant 120588 gt 0 then it holds that

Ψ (119905) = int1198911119901le120588(12058101+1205831)1199061

11990611990111198911minus1199011119901 119889119909+ int1198911119901gt120588(12058101+1205831)1199061

11990611990111198911minus1199011119901 119889119909le 120588 (12058101 + 1205831) int

1198911119901le120588(12058101+1205831)1199061119906119901+11 119891minus1199011119901 119889119909

+ (120588 (12058101 + 1205831))1minus119901 int1198911119901gt120588(12058101+1205831)1199061

1199061119889119909le 120588 (12058101 + 1205831) int

Ω119906119901+11 119891minus1199011119901 119889119909

+ (120588 (12058101 + 1205831))1minus119901 intΩ1199061119889119909

(62)

Hence

minus (12058101 + 1205831) intΩ119906119901+11 119891minus1199011119901 119889119909

le minus1120588Ψ (119905) + 120588minus119901 (12058101 + 1205831)1minus119901 intΩ1199061119889119909

(63)

Substituting (63) into (51) we have

1119901 minus 1 119889119889119909Ψ (119905) le ( 12058111 minus 120591 + 119901119901 minus 11205831 minus 1120588)Ψ (119905)+ 120588minus119901 (12058101 + 1205831)1minus119901 int

Ω1199061119889119909

(64)

Take 1120588 gt 1205811(1 minus 120591) + (119901(119901 minus 1))1205831 Noticing Lemma 5and initial date 11990610 isin 1198621(Ω) we apply the Gronwall Lemmato obtain

Ψ (119905) le max120588minus119901 (12058101 + 1205831)1minus119901

sdot ( 1120588 minus 12058111 minus 120591 minus 119901119901 minus 11205831)minus1 intΩ1199061119889119909 Ψ (0)

(65)

taking limits leads to

lim119901rarrinfin

Ψ1119901 (119905) le max 120588minus1 (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω) (66)

Since

Ψ (119905) = intΩ11990611990111198911minus1199011119901 (V) 119889119909 ge 1198911minus1199011119901 (V) int

Ω1199061199011119889119909 (67)

combining (66) and (67) and taking limits as 120588minus1 rarr 1205811(1 minus120591) + 1205831 it is found that (60) holds The same processes areused to prove (61) which ends the proof of Lemma 8

Remark 9 In the above arguments we used well such factLetΩ be bounded and regular domain ofR119899 If 119906119871119901(Ω) le 119862where119862 is some positive constant and independent of119901 then119906119871119901(Ω) rarr 119906119871infin(Ω) as 119901 rarr infin

To end the arguments of uniform boundedness of solu-tions it follows to prove that 119879lowast = 119879max and 119879max = +infin bycontradiction

As in Lemmas 4ndash8 we define 1 2 and V for the uniquesolutions in (0 max) to

1119905 = Δ1 minus nabla sdot (11205941 (V) nablaV) + 120583111198921 (1 2) 119909 isin Ω 119905 gt 0

2119905 = Δ2 minus nabla sdot (21205942 (V) nablaV) + 120583211990621198922 (1 2) 119909 isin Ω 119905 gt 0

V119905 = 120591ΔV + ℎ (1 2 V) 119909 isin Ω 119905 gt 01205971120597] = 1205972120597] = 120597V120597] = 0 119909 isin 120597Ω 119905 gt 0

1 (119909 0) = 11990610 (119909) 2 (119909 0) = 11990620 (119909) V (119909 0) = V0 (119909)

119909 isin Ω

(68)

where max is defined in the same fashion as (32) and thetruncation ℎ(1199061 1199062 V) is defined as follows

ℎ (1199061 1199062 V) fl ℎ (1199061 1199062 V) if V le Vℎ (1199061 1199062 V) if V gt V (69)

Then it is easy to see that100381710038171003817100381711003817100381710038171003817119871infin(Ω) le 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

100381710038171003817100381721003817100381710038171003817119871infin(Ω) le 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(70)

as far as V le VBy analogy with 119879lowast we define

lowast fl max if V le V in (0 max) V if V119871infin(Ωmax )

gt V (71)

Discrete Dynamics in Nature and Society 9

where V lt max satisfies that V119871infin(ΩV ) le V and

V119871infin(ΩV+120575) gt V for any 120575 gt 0 To prove lowast = max wesuppose

lowast lt max (72)

then ℎ(1 2 V) le ℎ(1199061 1199062 V) = ℎ(1199061 1199062 V) lt 0 It follows toeasily check that V is a strict upper-solution to

V119905 minus 120591ΔV = ℎ (1 2 V) in Ω times (0 lowast) 120597V120597] = 0 on 120597Ω times (0 lowast)

V (119909 0) = V0 (119909) in Ω(73)

This implies V(119909 119905) lt V forall119909 isin Ω 119905 isin (0 lowast) whichcontradicts (72) and proves lowast = max as well as max = infin

Thanks to V le V we show that (1 2 V) is also a solutionto (6) So we have

119879lowast = 119879max ge max = infin (74)

The proof of Theorem 3 is completed

Competing Interests

The author declares that they have no competing interests

Acknowledgments

Thiswork is supported by the ChinaNational Natural ScienceFoundation (nos 11361055 and 11261053) and the NaturalScience ResearchKey Project of University of Anhui Province(no KJ2015A251)

References

[1] J D Murray Mathematical Biology vol 19 Springer BerlinGermany 2nd edition 1993

[2] M Winkler ldquoBoundedness in the higher-dimensionalparabolic-parabolic chemotaxis system with logistic sourcerdquoCommunications in Partial Differential Equations vol 35 no 8pp 1516ndash1537 2010

[3] T Hillen and K J Painter ldquoA userrsquos guide to PDE models forchemotaxisrdquo Journal of Mathematical Biology vol 58 no 1-2pp 183ndash217 2009

[4] E F Keller and L A Segel ldquoInitiation of slimemold aggregationviewed as an instabilityrdquo Journal of Theoretical Biology vol 26no 3 pp 399ndash415 1970

[5] M Winkler ldquoAbsence of collapse in a parabolic chemo-taxis system with signal-dependent sensitivityrdquo MathematischeNachrichten vol 283 no 11 pp 1664ndash1673 2010

[6] J I Tello and M Winkler ldquoStabilization in a two-specieschemotaxis system with a logistic sourcerdquo Nonlinearity vol 25no 5 pp 1413ndash1425 2012

[7] M Negreanu and J I Tello ldquoAsymptotic stability of a twospecies chemotaxis systemwith non-diffusive chemoattractantrdquoJournal of Differential Equations vol 258 no 5 pp 1592ndash16172015

[8] Q Zhang and Y Li ldquoGlobal boundedness of solutions toa two-species chemotaxis systemrdquo Zeitschrift fur angewandteMathematik und Physik vol 66 no 1 pp 83ndash93 2015

[9] C Mu L Wang P Zheng and Q Zhang ldquoGlobal existenceand boundedness of classical solutions to a parabolic-parabolicchemotaxis systemrdquo Nonlinear Analysis Real World Applica-tions vol 14 no 3 pp 1634ndash1642 2013

[10] Y Tao ldquoBoundedness in a chemotaxis model with oxygenconsumption by bacteriardquo Journal ofMathematical Analysis andApplications vol 381 no 2 pp 521ndash529 2011

[11] D Horstmann and M Winkler ldquoBoundedness vs blow-up in achemotaxis systemrdquo Journal of Differential Equations vol 215no 1 pp 52ndash107 2005

[12] M Winkler ldquoAggregation vs global diffusive behavior in thehigher-dimensional Keller-Segel modelrdquo Journal of DifferentialEquations vol 248 no 12 pp 2889ndash2905 2010

[13] PQuittner andP Souplet Superlinear Parabolic Problems Blow-Up Global Existence and Steady States Birkhauser AdvancedTexts Birkhauser Basel Switzerland 2007

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Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

3 Global Existence of Solutions

We will first be devoted to dealing with local-in-time exis-tence and uniqueness of a nonnegative solution for (1) Thecorresponding conclusions are written by Lemma 4

31 Local Existence of Solutions

Lemma 4 Given positive initial date (11990610 11990620 V0) isin (1198621(Ω))3and the same assumptions as inTheorem 3 there exists a smallenough time 119879 gt 0 and a unique triple of nonnegativefunctions (1199061 1199062 V) isin (119871119901([0 119879]1198822119901(Ω)) cap 1198821119901([0 119879]119871119901(Ω)))3 (for 119901 gt 119899) such that (1199061 1199062 V) is a solution of (1)in Ω times (0 119879) Moreover we have V ge V

Proof Introduce the change of variables 1 and 2 given by

1199061 = 1198911 (V) 11199062 = 1198912 (V) 2 (20)

where the function 119891119894(V) is defined by

119891119894 (V) = exp 11 minus 120591 intV

V120594119894 (119904) 119889119904 119894 = 1 2 (21)

A direct calculation yields

1199061119905 = 11989110158401 (V) (120591ΔV + ℎ (1199061 1199062 V)) 1+ 1198911 (V) 1119905

Δ1199061 = 119891101584010158401 (V) 1 |nablaV|2 + 211989110158401 (V) nabla1 sdot nablaV+ 11989110158401 (V) 1ΔV + 1198911 (V) Δ1

nabla sdot (11990611205941 (V) nablaV) = 11989110158401 (V) 11205941 (V) |nablaV|2+ 1198911 (V) 1205941 (V) nabla1 sdot nablaV+ 1198911 (V) 112059410158401 (V) |nablaV|2+ 1198911 (V) 11205941 (V) ΔV

(22)

similarly

1199062119905 = 11989110158402 (V) (120591ΔV + ℎ (1199061 1199062 V)) 2+ 1198912 (V) 2119905

Δ1199062 = 119891101584010158402 (V) 2 |nablaV|2 + 211989110158402 (V) nabla2 sdot nablaV+ 11989110158402 (V) 2ΔV + 1198912 (V) Δ2

nabla sdot (11990621205942 (V) nablaV) = 11989110158402 (V) 21205942 (V) |nablaV|2+ 1198912 (V) 1205942 (V) nabla2 sdot nablaV+ 1198912 (V) 212059410158402 (V) |nablaV|2+ 1198912 (V) 21205942 (V) ΔV

(23)

where

1198911015840119894 (V) = 120594119894 (V)1 minus 120591119891119894 (V) 11989110158401015840119894 (V) = ( 1205942119894 (V)(1 minus 120591)2 +

1205941015840119894 (V)1 minus 120591 )119891119894 (V) 119894 = 1 2

(24)

Substitute (20) (22) and (23) into (1) and denote that

L119894 (V) 119894 = 119894119905 minus Δ119894 minus 1 + 1205911 minus 120591120594119894 (V) nablaV sdot nabla119894119866119894 (1 2 V) = minus119909119894 (V)1 minus 120591 ℎ (1198911 (V) 1 1198912 (V) 2 V)+ 120583119894119892119894 (1198911 (V) 1 1198912 (V) 2)

(25)

for 119894 = 1 2Then system (1) becomes

L1 (V) 1= 11198661 (1 2 V)

+ 1 120591(1 minus 120591)2 (12059421 (V) + (1 minus 120591) 12059410158401 (V)) |nablaV|2

119909 isin Ω 119905 gt 0L2 (V) 2

= 21198662 (1 2 V)+ 2 120591

(1 minus 120591)2 (12059422 (V) + (1 minus 120591) 12059410158402 (V)) |nablaV|2 119909 isin Ω 119905 gt 0

V119905 = 120591ΔV + ℎ (1198911 (V) 1 1198912 (V) 2 V) 119909 isin Ω 119905 gt 0

(26)

with boundary conditions 1205971120597] = 1205972120597] = 120597V120597] = 0119909 isin 120597Ω 119905 gt 0 and initial conditions 1(119909 0) fl 10(119909) =11990610(119909)1198911(V0(119909)) 2(119909 0) fl 20(119909) = 11990620(119909)1198912(V0(119909))V(119909 0) = V0(119909) 119909 isin Ω

For sufficiently small 119879 gt 0 in the space [119871119901([0 119879]1198822119901(Ω)) cap 1198821119901([0 119879] 119871119901(Ω))]2 (for 119901 gt 119899) we implementa fixed point argument Taking V isin 119862([0 119879] 1198621(Ω)) satisfyingV isin (V V) and |nablaV| lt 119862 we define 1 2 in (0 119879) as the uniquesolution to the questionL1 (V) 1

= 11198661 (1 2 V)+ 1 120591

(1 minus 120591)2 (12059421 (V) + (1 minus 120591) 12059410158401 (V)) |nablaV|2in Ω times (0 119879)

L2 (V) 2= 21198662 (1 2 V)

+ 2 120591(1 minus 120591)2 (12059422 (V) + (1 minus 120591) 12059410158402 (V)) |nablaV|2

in Ω times (0 119879)

(27)

Discrete Dynamics in Nature and Society 5

for the details we refer the reader to [13 Remark 483]Noticethat the right-hand side terms of (27) are the multiplicativefor 1 and 2 applying the maximum principle to verify thatboth 1 and 2 are nonnegative

Since condition (11) we easily see that the regular func-tions 1 and 2 satisfy

L1 (V) 1 le 11198661 (1 2 V) L2 (V) 2 le 21198662 (1 2 V) (28)

and the right-hand side terms of (27) are quasi-decreasedlooking upon the lower-solution 1199061 = 1199062 = 0 one mayconstruct upper-solutions 1199061 and 1199062 such that

0 le 1199061 le 11990610 le 1199062 le 1199062119905 lt 119879

(29)

It follows that we apply Theorem 21 of [7] to get a uniquenonnegative solution

We solve the parabolic equation

V119905 = 120591ΔV + ℎ (1198911 (V) 1 1198912 (V) 2 V) (30)

Thanks to the essential estimations of parabolic equations andthe embedding theorems we apply the Schauder fixed pointtheorem to obtain the local existence of solution V(119909 119905) Thesmoothness of ℎ(1198911(V)1 1198912(V)2 V) ensures the uniquenessof solution V(119909 119905)

To prove V ge V in view of assumption (8) and themonotone of ℎ(1199061 1199062 V) we have 0 le ℎ(0 0 V) le ℎ(1199061 1199062 V)It follows that V is a lower-solution for the equation

V119905 minus 120591ΔV = ℎ (1199061 1199062 V) in Ω times (0 119879) 120597V120597] = 0 on 120597Ω times (0 119879)

V (119909 0) = V0 (119909) in Ω(31)

The proof of Lemma 4 is completed

About the above solution (1199061 1199062 V) we have the followingextensibility criterionThe solution is extended to the interval(0 119879max) where 119879max has the following property

If 119879max lt infinthen lim sup

119905rarr119879max

(10038171003817100381710038171199061 (sdot 119905)1003817100381710038171003817119871infin(Ω) + 10038171003817100381710038171199062 (sdot 119905)1003817100381710038171003817119871infin(Ω)+ V (sdot 119905)119871infin(Ω)) = infin

(32)

32 Uniform Boundedness of Solutions To obtain some a pri-ori estimates we need some technical lemmasThe following1198711-estimate of solution is first given

Lemma 5 For all 119905 isin (0 119879119898119886119909) the solutions to (1) satisfy thefollowing estimates

1003817100381710038171003817119906110038171003817100381710038171198711(Ω) le max |Ω| 10038171003817100381710038171199061010038171003817100381710038171198711(Ω) le 1198721003817100381710038171003817119906210038171003817100381710038171198711(Ω) le max |Ω| 10038171003817100381710038171199062010038171003817100381710038171198711(Ω) le 119872V1198711(Ω) le max 1003817100381710038171003817V010038171003817100381710038171198711(Ω) 2119872120574minus1

(33)

where119872 = max|Ω| 119906101198711(Ω) 119906201198711(Ω)Proof Integrating the first equation of (1) overΩ we have

119889119889119905 intΩ 1199061119889119909 = 1205831 intΩ(1199061 minus 11990621 minus 119886111990611199062) 119889119909

le 1205831 intΩ(1199061 minus 11990621) 119889119909

(34)

Using Cauchy inequality yields

119889119889119905 intΩ 1199061119889119909 le 1205831 intΩ(1 minus 1199061) 119889119909

le 1205831 |Ω| minus 1205831 intΩ1199061119889119909

(35)

By the Gronwall Lemma we derive

10038171003817100381710038171199061 (119909 119905)10038171003817100381710038171198711(Ω) le max |Ω| 100381710038171003817100381711990610 (119909)10038171003817100381710038171198711(Ω) le 119872 (36)

a similar estimate holds for 1199062 which leads to

10038171003817100381710038171199062 (119909 119905)10038171003817100381710038171198711(Ω) le max |Ω| 100381710038171003817100381711990620 (119909)10038171003817100381710038171198711(Ω) le 119872 (37)

Integrating the third equation of (1) overΩ we have

119889119889119905 intΩ V 119889119909 = intΩ(minus120574V + 1199061 + 1199062) 119889119909

= minus120574intΩV 119889119909 + int

Ω(1199061 + 1199062) 119889119909

le minus120574intΩV 119889119909 + 2119872

(38)

it implies

V1198711(Ω) le max 1003817100381710038171003817V010038171003817100381710038171198711(Ω) 2119872120574minus1 (39)

This proves the lemma

6 Discrete Dynamics in Nature and Society

Lemma 6 Letting 119901 gt 1 and under assumption (11) then thefollowing estimates hold

119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909le 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990111205941 (V) 1198911minus1199011119901 ℎ (1199061 1199062 V) 119889119909+ 1199011205831 int

Ω11990611990111198911minus1199011119901 1198921 (1199061 1199062) 119889119909

(40)

119889119889119905 intΩ 11990611990121198911minus1199012119901 119889119909le 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990121205942 (V) 1198911minus1199012119901 ℎ (1199061 1199062 V) 119889119909+ 1199011205832 int

Ω11990611990121198911minus1199012119901 1198922 (1199061 1199062) 119889119909

(41)

where 119891119894119901(V) is defined by

119891119894119901 (V) = exp( 11 minus 120591 ((119901 minus 1) 119901) intV

V120594119894 (119904) 119889119904)

119894 = 1 2(42)

Proof It is easy to check

11989110158401119901 (V) = 11 minus 120591 ((119901 minus 1) 119901)1205941 (V) 1198911119901 (V) 11989110158402119901 (V) = 11 minus 120591 ((119901 minus 1) 119901)1205942 (V) 1198912119901 (V)

(43)

Then for any 119901 gt 1 we have119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909 = 119901int

Ω119906119901minus11 11990611199051198911minus1199011119901 119889119909 + (1 minus 119901)

sdot intΩ1199061199011119891minus1199011119901 11989110158401119901V119905119889119909 = 119901int

Ω119906119901minus11 1198911minus1199011119901 [Δ1199061 minus nabla

sdot (11990611205941 (V) nablaV) + 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1)11990611205941 (V) ΔV]119889119909

+ 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990111198911minus1199011119901 1205941 (V)

sdot ℎ (1199061 1199062 V) 119889119909+ 1199011205831 int

Ω11990611990111198911minus1199011119901 1198921 (1199061 1199062) 119889119909 fl 1198681 + 1198682 + 1198683

(44)

Clearly if 1198681 le 0 then (40) is a consequence of (44) We inthe following verify that the result holds

In fact

1198681 = 119901intΩ119906119901minus11 1198911minus1199011119901 [Δ1199061 minus nabla sdot (11990611205941 (V) nablaV)

+ 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1) (nabla sdot (11990611205941 (V) nablaV) minus nabla (11990611205941 (V))

sdot nablaV)] 119889119909 = 119901intΩ119906119901minus11 1198911minus1199011119901 [nabla

sdot (1198911119901nabla (1199061119891minus11119901 )) minus 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1) (1205941 (V) nabla1199061 sdot nablaV

+ 119906112059410158401 (V) |nablaV|2)] 119889119909 = 119901 (1 minus 119901)

sdot intΩ119906119901minus21 1198912minus1199011119901 [1198911119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162

minus 1205911199061 (1205941 (V) nabla1199061 sdot nablaV + 119906112059410158401 (V) |nablaV|2)1198911119901 (119901 minus 120591 (119901 minus 1)) ] 119889119909= 119901 (1 minus 119901)int

Ω119906119901minus21 1198913minus1199011119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162 119889119909

+ 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ 119906119901minus11 1198911minus1199011119901 1205941 (V) nabla1199061 sdot nablaV 119889119909

+ 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ 11990611990111198911minus1199011119901 12059410158401 (V) |nablaV|2 119889119909 fl 1198691

+ 1198692 + 1198693

(45)

Since nabla1199061 = nabla(1198911119901(1199061119891minus11119901 )) multiplying both sides by1205941(V)nablaV we get1205941 (V) nabla1199061 sdot nablaV

= 11989111199011205941 (V) nabla (1199061119891minus11119901 ) sdot nablaV+ 11 minus 120591 ((119901 minus 1) 119901)119906112059421 (V) |nablaV|2

(46)

and by 120576-Youngrsquos inequality1205941 (V) nabla (1199061119891minus11119901 ) sdot nablaV

le 119901 minus 120591 (119901 minus 1)120591

1198911119901119906110038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162

+ 120591119901 minus 120591 (119901 minus 1) 1199061119891111990112059421 (V) |nablaV|2

(47)

Combining (46) and (47) and substituting it into 1198692 we derive1198692 le 119901 (119901 minus 1)int

Ω119906119901minus21 1198913minus1199011119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162 119889119909

+ 120591119901 (119901 minus 1) (120591 + 119901)(119901 minus 120591 (119901 minus 1))2 int

Ω11990611990111198911minus1199011119901 12059421 (V) |nablaV|2 119889119909

(48)

Discrete Dynamics in Nature and Society 7

Inserting (48) into (45) yields

1198681 le 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ (

120591 + 119901(119901 minus 120591 (119901 minus 1))12059421 (V)

+ 12059410158401 (V)) 11990611990111198911minus1199011119901 |nablaV|2 119889119909 = 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1)

sdot intΩ[ 1(1 minus 120591) (1 minus 1205912120591 + (1 minus 120591) 119901)12059421 (V)

+ 12059410158401 (V)] 11990611990111198911minus1199011119901 |nablaV|2 119889119909 le 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1)

sdot intΩ( 1(1 minus 120591)12059421 (V) + 12059410158401 (V)) 11990611990111198911minus1199011119901 |nablaV|2 119889119909

(49)

we deduce from assumption (11) that 1198681 le 0 for any 119901 gt 1and the proof ends In the same way we achieve (41) Thiscompletes the proof of Lemma 6

In view of V isin 119862((0 119879max) 1198621(Ω)) and initial date V ltV0 lt V we consider 119879lowast gt 0 defined by

119879lowast fl 119879max if V le V in (0 119879max) 119879V if V119871infin(Ω119879max )

gt V (50)

where 119879V lt 119879max satisfies that V119871infin(Ω119879V ) le V andV119871infin(Ω119879V+120575) gt V for any 120575 gt 0Lemma7 Letting119901 gt 1 and for any 119905 lt 119879lowast then the solutionsto (1) satisfy

1119901 minus 1 119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909le minus (12058101 + 1205831) int

Ω119906119901+11 119891minus1199011119901 119889119909

+ ( 12058111 minus 120591 + 119901119901 minus 11205831)intΩ11990611990111198911minus1199011119901 119889119909

(51)

1119901 minus 1 119889119889119905 intΩ 11990611990121198911minus1199012119901 119889119909le minus (12058102 + 1205832) int

Ω119906119901+12 119891minus1199012119901 119889119909

+ ( 12058121 minus 120591 + 119901119901 minus 11205832)intΩ11990611990121198911minus1199012119901 119889119909

(52)

Proof In order to prove Lemma 7 we consider the terms 1198682and 1198683 appearing in the proof of Lemma 6

By the nonnegativity for 1199061 1199062 and assumption (6) weobtain

minusℎ (1199061 1199062 V) le minus1199061 + 12058111205941 (V) (53)

Multiplying (53) by 11990611990111198911minus1199011119901 1205941(V) and integrating overΩ yield

minus intΩ11990611990111198911minus1199011119901 1205941 (V) ℎ (1199061 1199062 V) 119889119909

le minusintΩ119906119901+11 1198911minus1199011119901 1205941 (V) 119889119909 + 1205811 int

Ω11990611990111198911minus1199011119901 119889119909

(54)

By the monotonic property of exponential function andassumption (7) we have

0 lt 12058101 le 1205941 (V) exp(intV

V1205941 (119904) 119889119904) le 1205941 (V) 1198911119901 (V) (55)

It follows that (54) becomes

minus intΩ11990611990111198911minus1199011119901 1205941 (V) ℎ (1199061 1199062 V) 119889119909

le minus12058101 intΩ119906119901+11 119891minus1199011119901 119889119909 + 1205811 int

Ω11990611990111198911minus1199011119901 119889119909

(56)

Multiplying (56) by (119901 minus 1)(1 minus 120591((119901 minus 1)119901)) we get1198682 le 119901 minus 1

1 minus 120591 ((119901 minus 1) 119901) (minus12058101 intΩ 119906119901+11 119891minus1199011119901 119889119909+ 1205811 int

Ω11990611990111198911minus1199011119901 119889119909) le (119901 minus 1)

sdot (minus12058101 intΩ119906119901+11 119891minus1199011119901 119889119909 + 12058111 minus 120591 int

Ω11990611990111198911minus1199011119901 119889119909)

(57)

For the term 1198683 we easily achieve1198683 le 1199011205831 (int

Ω11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

le (119901 minus 1) 1205831 (intΩ11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

+ 1205831 intΩ11990611990111198911minus1199011119901 119889119909

(58)

Combining (44) (57) and (58) leads to

119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909 le (119901 minus 1)sdot (minus12058101 int

Ω119906119901+11 119891minus1199011119901 119889119909 + 12058111 minus 120591 int

Ω11990611990111198911minus1199011119901 119889119909)

+ (119901 minus 1) 1205831 (intΩ11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

+ 1205831 intΩ11990611990111198911minus1199011119901 119889119909

(59)

which implies that (51) holds The same arguments are usedto achieve (52) The proof ends

To establish the uniform boundedness of solutions weapply an iterative method based on the 119871119901-norm of solutionsinspired by the Moser-Alikakos type iterations (see [10 13])

8 Discrete Dynamics in Nature and Society

Lemma 8 For any 119905 lt 119879lowast then the solutions to (1) arebounded by

100381710038171003817100381711990611003817100381710038171003817119871infin(Ω) le 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

(60)

100381710038171003817100381711990621003817100381710038171003817119871infin(Ω) le 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(61)

Proof DenoteintΩ 11990611990111198911minus1199011119901 119889119909 byΨ(119905) Taking arbitrary positiveconstant 120588 gt 0 then it holds that

Ψ (119905) = int1198911119901le120588(12058101+1205831)1199061

11990611990111198911minus1199011119901 119889119909+ int1198911119901gt120588(12058101+1205831)1199061

11990611990111198911minus1199011119901 119889119909le 120588 (12058101 + 1205831) int

1198911119901le120588(12058101+1205831)1199061119906119901+11 119891minus1199011119901 119889119909

+ (120588 (12058101 + 1205831))1minus119901 int1198911119901gt120588(12058101+1205831)1199061

1199061119889119909le 120588 (12058101 + 1205831) int

Ω119906119901+11 119891minus1199011119901 119889119909

+ (120588 (12058101 + 1205831))1minus119901 intΩ1199061119889119909

(62)

Hence

minus (12058101 + 1205831) intΩ119906119901+11 119891minus1199011119901 119889119909

le minus1120588Ψ (119905) + 120588minus119901 (12058101 + 1205831)1minus119901 intΩ1199061119889119909

(63)

Substituting (63) into (51) we have

1119901 minus 1 119889119889119909Ψ (119905) le ( 12058111 minus 120591 + 119901119901 minus 11205831 minus 1120588)Ψ (119905)+ 120588minus119901 (12058101 + 1205831)1minus119901 int

Ω1199061119889119909

(64)

Take 1120588 gt 1205811(1 minus 120591) + (119901(119901 minus 1))1205831 Noticing Lemma 5and initial date 11990610 isin 1198621(Ω) we apply the Gronwall Lemmato obtain

Ψ (119905) le max120588minus119901 (12058101 + 1205831)1minus119901

sdot ( 1120588 minus 12058111 minus 120591 minus 119901119901 minus 11205831)minus1 intΩ1199061119889119909 Ψ (0)

(65)

taking limits leads to

lim119901rarrinfin

Ψ1119901 (119905) le max 120588minus1 (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω) (66)

Since

Ψ (119905) = intΩ11990611990111198911minus1199011119901 (V) 119889119909 ge 1198911minus1199011119901 (V) int

Ω1199061199011119889119909 (67)

combining (66) and (67) and taking limits as 120588minus1 rarr 1205811(1 minus120591) + 1205831 it is found that (60) holds The same processes areused to prove (61) which ends the proof of Lemma 8

Remark 9 In the above arguments we used well such factLetΩ be bounded and regular domain ofR119899 If 119906119871119901(Ω) le 119862where119862 is some positive constant and independent of119901 then119906119871119901(Ω) rarr 119906119871infin(Ω) as 119901 rarr infin

To end the arguments of uniform boundedness of solu-tions it follows to prove that 119879lowast = 119879max and 119879max = +infin bycontradiction

As in Lemmas 4ndash8 we define 1 2 and V for the uniquesolutions in (0 max) to

1119905 = Δ1 minus nabla sdot (11205941 (V) nablaV) + 120583111198921 (1 2) 119909 isin Ω 119905 gt 0

2119905 = Δ2 minus nabla sdot (21205942 (V) nablaV) + 120583211990621198922 (1 2) 119909 isin Ω 119905 gt 0

V119905 = 120591ΔV + ℎ (1 2 V) 119909 isin Ω 119905 gt 01205971120597] = 1205972120597] = 120597V120597] = 0 119909 isin 120597Ω 119905 gt 0

1 (119909 0) = 11990610 (119909) 2 (119909 0) = 11990620 (119909) V (119909 0) = V0 (119909)

119909 isin Ω

(68)

where max is defined in the same fashion as (32) and thetruncation ℎ(1199061 1199062 V) is defined as follows

ℎ (1199061 1199062 V) fl ℎ (1199061 1199062 V) if V le Vℎ (1199061 1199062 V) if V gt V (69)

Then it is easy to see that100381710038171003817100381711003817100381710038171003817119871infin(Ω) le 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

100381710038171003817100381721003817100381710038171003817119871infin(Ω) le 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(70)

as far as V le VBy analogy with 119879lowast we define

lowast fl max if V le V in (0 max) V if V119871infin(Ωmax )

gt V (71)

Discrete Dynamics in Nature and Society 9

where V lt max satisfies that V119871infin(ΩV ) le V and

V119871infin(ΩV+120575) gt V for any 120575 gt 0 To prove lowast = max wesuppose

lowast lt max (72)

then ℎ(1 2 V) le ℎ(1199061 1199062 V) = ℎ(1199061 1199062 V) lt 0 It follows toeasily check that V is a strict upper-solution to

V119905 minus 120591ΔV = ℎ (1 2 V) in Ω times (0 lowast) 120597V120597] = 0 on 120597Ω times (0 lowast)

V (119909 0) = V0 (119909) in Ω(73)

This implies V(119909 119905) lt V forall119909 isin Ω 119905 isin (0 lowast) whichcontradicts (72) and proves lowast = max as well as max = infin

Thanks to V le V we show that (1 2 V) is also a solutionto (6) So we have

119879lowast = 119879max ge max = infin (74)

The proof of Theorem 3 is completed

Competing Interests

The author declares that they have no competing interests

Acknowledgments

Thiswork is supported by the ChinaNational Natural ScienceFoundation (nos 11361055 and 11261053) and the NaturalScience ResearchKey Project of University of Anhui Province(no KJ2015A251)

References

[1] J D Murray Mathematical Biology vol 19 Springer BerlinGermany 2nd edition 1993

[2] M Winkler ldquoBoundedness in the higher-dimensionalparabolic-parabolic chemotaxis system with logistic sourcerdquoCommunications in Partial Differential Equations vol 35 no 8pp 1516ndash1537 2010

[3] T Hillen and K J Painter ldquoA userrsquos guide to PDE models forchemotaxisrdquo Journal of Mathematical Biology vol 58 no 1-2pp 183ndash217 2009

[4] E F Keller and L A Segel ldquoInitiation of slimemold aggregationviewed as an instabilityrdquo Journal of Theoretical Biology vol 26no 3 pp 399ndash415 1970

[5] M Winkler ldquoAbsence of collapse in a parabolic chemo-taxis system with signal-dependent sensitivityrdquo MathematischeNachrichten vol 283 no 11 pp 1664ndash1673 2010

[6] J I Tello and M Winkler ldquoStabilization in a two-specieschemotaxis system with a logistic sourcerdquo Nonlinearity vol 25no 5 pp 1413ndash1425 2012

[7] M Negreanu and J I Tello ldquoAsymptotic stability of a twospecies chemotaxis systemwith non-diffusive chemoattractantrdquoJournal of Differential Equations vol 258 no 5 pp 1592ndash16172015

[8] Q Zhang and Y Li ldquoGlobal boundedness of solutions toa two-species chemotaxis systemrdquo Zeitschrift fur angewandteMathematik und Physik vol 66 no 1 pp 83ndash93 2015

[9] C Mu L Wang P Zheng and Q Zhang ldquoGlobal existenceand boundedness of classical solutions to a parabolic-parabolicchemotaxis systemrdquo Nonlinear Analysis Real World Applica-tions vol 14 no 3 pp 1634ndash1642 2013

[10] Y Tao ldquoBoundedness in a chemotaxis model with oxygenconsumption by bacteriardquo Journal ofMathematical Analysis andApplications vol 381 no 2 pp 521ndash529 2011

[11] D Horstmann and M Winkler ldquoBoundedness vs blow-up in achemotaxis systemrdquo Journal of Differential Equations vol 215no 1 pp 52ndash107 2005

[12] M Winkler ldquoAggregation vs global diffusive behavior in thehigher-dimensional Keller-Segel modelrdquo Journal of DifferentialEquations vol 248 no 12 pp 2889ndash2905 2010

[13] PQuittner andP Souplet Superlinear Parabolic Problems Blow-Up Global Existence and Steady States Birkhauser AdvancedTexts Birkhauser Basel Switzerland 2007

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Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

for the details we refer the reader to [13 Remark 483]Noticethat the right-hand side terms of (27) are the multiplicativefor 1 and 2 applying the maximum principle to verify thatboth 1 and 2 are nonnegative

Since condition (11) we easily see that the regular func-tions 1 and 2 satisfy

L1 (V) 1 le 11198661 (1 2 V) L2 (V) 2 le 21198662 (1 2 V) (28)

and the right-hand side terms of (27) are quasi-decreasedlooking upon the lower-solution 1199061 = 1199062 = 0 one mayconstruct upper-solutions 1199061 and 1199062 such that

0 le 1199061 le 11990610 le 1199062 le 1199062119905 lt 119879

(29)

It follows that we apply Theorem 21 of [7] to get a uniquenonnegative solution

We solve the parabolic equation

V119905 = 120591ΔV + ℎ (1198911 (V) 1 1198912 (V) 2 V) (30)

Thanks to the essential estimations of parabolic equations andthe embedding theorems we apply the Schauder fixed pointtheorem to obtain the local existence of solution V(119909 119905) Thesmoothness of ℎ(1198911(V)1 1198912(V)2 V) ensures the uniquenessof solution V(119909 119905)

To prove V ge V in view of assumption (8) and themonotone of ℎ(1199061 1199062 V) we have 0 le ℎ(0 0 V) le ℎ(1199061 1199062 V)It follows that V is a lower-solution for the equation

V119905 minus 120591ΔV = ℎ (1199061 1199062 V) in Ω times (0 119879) 120597V120597] = 0 on 120597Ω times (0 119879)

V (119909 0) = V0 (119909) in Ω(31)

The proof of Lemma 4 is completed

About the above solution (1199061 1199062 V) we have the followingextensibility criterionThe solution is extended to the interval(0 119879max) where 119879max has the following property

If 119879max lt infinthen lim sup

119905rarr119879max

(10038171003817100381710038171199061 (sdot 119905)1003817100381710038171003817119871infin(Ω) + 10038171003817100381710038171199062 (sdot 119905)1003817100381710038171003817119871infin(Ω)+ V (sdot 119905)119871infin(Ω)) = infin

(32)

32 Uniform Boundedness of Solutions To obtain some a pri-ori estimates we need some technical lemmasThe following1198711-estimate of solution is first given

Lemma 5 For all 119905 isin (0 119879119898119886119909) the solutions to (1) satisfy thefollowing estimates

1003817100381710038171003817119906110038171003817100381710038171198711(Ω) le max |Ω| 10038171003817100381710038171199061010038171003817100381710038171198711(Ω) le 1198721003817100381710038171003817119906210038171003817100381710038171198711(Ω) le max |Ω| 10038171003817100381710038171199062010038171003817100381710038171198711(Ω) le 119872V1198711(Ω) le max 1003817100381710038171003817V010038171003817100381710038171198711(Ω) 2119872120574minus1

(33)

where119872 = max|Ω| 119906101198711(Ω) 119906201198711(Ω)Proof Integrating the first equation of (1) overΩ we have

119889119889119905 intΩ 1199061119889119909 = 1205831 intΩ(1199061 minus 11990621 minus 119886111990611199062) 119889119909

le 1205831 intΩ(1199061 minus 11990621) 119889119909

(34)

Using Cauchy inequality yields

119889119889119905 intΩ 1199061119889119909 le 1205831 intΩ(1 minus 1199061) 119889119909

le 1205831 |Ω| minus 1205831 intΩ1199061119889119909

(35)

By the Gronwall Lemma we derive

10038171003817100381710038171199061 (119909 119905)10038171003817100381710038171198711(Ω) le max |Ω| 100381710038171003817100381711990610 (119909)10038171003817100381710038171198711(Ω) le 119872 (36)

a similar estimate holds for 1199062 which leads to

10038171003817100381710038171199062 (119909 119905)10038171003817100381710038171198711(Ω) le max |Ω| 100381710038171003817100381711990620 (119909)10038171003817100381710038171198711(Ω) le 119872 (37)

Integrating the third equation of (1) overΩ we have

119889119889119905 intΩ V 119889119909 = intΩ(minus120574V + 1199061 + 1199062) 119889119909

= minus120574intΩV 119889119909 + int

Ω(1199061 + 1199062) 119889119909

le minus120574intΩV 119889119909 + 2119872

(38)

it implies

V1198711(Ω) le max 1003817100381710038171003817V010038171003817100381710038171198711(Ω) 2119872120574minus1 (39)

This proves the lemma

6 Discrete Dynamics in Nature and Society

Lemma 6 Letting 119901 gt 1 and under assumption (11) then thefollowing estimates hold

119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909le 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990111205941 (V) 1198911minus1199011119901 ℎ (1199061 1199062 V) 119889119909+ 1199011205831 int

Ω11990611990111198911minus1199011119901 1198921 (1199061 1199062) 119889119909

(40)

119889119889119905 intΩ 11990611990121198911minus1199012119901 119889119909le 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990121205942 (V) 1198911minus1199012119901 ℎ (1199061 1199062 V) 119889119909+ 1199011205832 int

Ω11990611990121198911minus1199012119901 1198922 (1199061 1199062) 119889119909

(41)

where 119891119894119901(V) is defined by

119891119894119901 (V) = exp( 11 minus 120591 ((119901 minus 1) 119901) intV

V120594119894 (119904) 119889119904)

119894 = 1 2(42)

Proof It is easy to check

11989110158401119901 (V) = 11 minus 120591 ((119901 minus 1) 119901)1205941 (V) 1198911119901 (V) 11989110158402119901 (V) = 11 minus 120591 ((119901 minus 1) 119901)1205942 (V) 1198912119901 (V)

(43)

Then for any 119901 gt 1 we have119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909 = 119901int

Ω119906119901minus11 11990611199051198911minus1199011119901 119889119909 + (1 minus 119901)

sdot intΩ1199061199011119891minus1199011119901 11989110158401119901V119905119889119909 = 119901int

Ω119906119901minus11 1198911minus1199011119901 [Δ1199061 minus nabla

sdot (11990611205941 (V) nablaV) + 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1)11990611205941 (V) ΔV]119889119909

+ 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990111198911minus1199011119901 1205941 (V)

sdot ℎ (1199061 1199062 V) 119889119909+ 1199011205831 int

Ω11990611990111198911minus1199011119901 1198921 (1199061 1199062) 119889119909 fl 1198681 + 1198682 + 1198683

(44)

Clearly if 1198681 le 0 then (40) is a consequence of (44) We inthe following verify that the result holds

In fact

1198681 = 119901intΩ119906119901minus11 1198911minus1199011119901 [Δ1199061 minus nabla sdot (11990611205941 (V) nablaV)

+ 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1) (nabla sdot (11990611205941 (V) nablaV) minus nabla (11990611205941 (V))

sdot nablaV)] 119889119909 = 119901intΩ119906119901minus11 1198911minus1199011119901 [nabla

sdot (1198911119901nabla (1199061119891minus11119901 )) minus 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1) (1205941 (V) nabla1199061 sdot nablaV

+ 119906112059410158401 (V) |nablaV|2)] 119889119909 = 119901 (1 minus 119901)

sdot intΩ119906119901minus21 1198912minus1199011119901 [1198911119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162

minus 1205911199061 (1205941 (V) nabla1199061 sdot nablaV + 119906112059410158401 (V) |nablaV|2)1198911119901 (119901 minus 120591 (119901 minus 1)) ] 119889119909= 119901 (1 minus 119901)int

Ω119906119901minus21 1198913minus1199011119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162 119889119909

+ 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ 119906119901minus11 1198911minus1199011119901 1205941 (V) nabla1199061 sdot nablaV 119889119909

+ 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ 11990611990111198911minus1199011119901 12059410158401 (V) |nablaV|2 119889119909 fl 1198691

+ 1198692 + 1198693

(45)

Since nabla1199061 = nabla(1198911119901(1199061119891minus11119901 )) multiplying both sides by1205941(V)nablaV we get1205941 (V) nabla1199061 sdot nablaV

= 11989111199011205941 (V) nabla (1199061119891minus11119901 ) sdot nablaV+ 11 minus 120591 ((119901 minus 1) 119901)119906112059421 (V) |nablaV|2

(46)

and by 120576-Youngrsquos inequality1205941 (V) nabla (1199061119891minus11119901 ) sdot nablaV

le 119901 minus 120591 (119901 minus 1)120591

1198911119901119906110038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162

+ 120591119901 minus 120591 (119901 minus 1) 1199061119891111990112059421 (V) |nablaV|2

(47)

Combining (46) and (47) and substituting it into 1198692 we derive1198692 le 119901 (119901 minus 1)int

Ω119906119901minus21 1198913minus1199011119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162 119889119909

+ 120591119901 (119901 minus 1) (120591 + 119901)(119901 minus 120591 (119901 minus 1))2 int

Ω11990611990111198911minus1199011119901 12059421 (V) |nablaV|2 119889119909

(48)

Discrete Dynamics in Nature and Society 7

Inserting (48) into (45) yields

1198681 le 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ (

120591 + 119901(119901 minus 120591 (119901 minus 1))12059421 (V)

+ 12059410158401 (V)) 11990611990111198911minus1199011119901 |nablaV|2 119889119909 = 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1)

sdot intΩ[ 1(1 minus 120591) (1 minus 1205912120591 + (1 minus 120591) 119901)12059421 (V)

+ 12059410158401 (V)] 11990611990111198911minus1199011119901 |nablaV|2 119889119909 le 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1)

sdot intΩ( 1(1 minus 120591)12059421 (V) + 12059410158401 (V)) 11990611990111198911minus1199011119901 |nablaV|2 119889119909

(49)

we deduce from assumption (11) that 1198681 le 0 for any 119901 gt 1and the proof ends In the same way we achieve (41) Thiscompletes the proof of Lemma 6

In view of V isin 119862((0 119879max) 1198621(Ω)) and initial date V ltV0 lt V we consider 119879lowast gt 0 defined by

119879lowast fl 119879max if V le V in (0 119879max) 119879V if V119871infin(Ω119879max )

gt V (50)

where 119879V lt 119879max satisfies that V119871infin(Ω119879V ) le V andV119871infin(Ω119879V+120575) gt V for any 120575 gt 0Lemma7 Letting119901 gt 1 and for any 119905 lt 119879lowast then the solutionsto (1) satisfy

1119901 minus 1 119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909le minus (12058101 + 1205831) int

Ω119906119901+11 119891minus1199011119901 119889119909

+ ( 12058111 minus 120591 + 119901119901 minus 11205831)intΩ11990611990111198911minus1199011119901 119889119909

(51)

1119901 minus 1 119889119889119905 intΩ 11990611990121198911minus1199012119901 119889119909le minus (12058102 + 1205832) int

Ω119906119901+12 119891minus1199012119901 119889119909

+ ( 12058121 minus 120591 + 119901119901 minus 11205832)intΩ11990611990121198911minus1199012119901 119889119909

(52)

Proof In order to prove Lemma 7 we consider the terms 1198682and 1198683 appearing in the proof of Lemma 6

By the nonnegativity for 1199061 1199062 and assumption (6) weobtain

minusℎ (1199061 1199062 V) le minus1199061 + 12058111205941 (V) (53)

Multiplying (53) by 11990611990111198911minus1199011119901 1205941(V) and integrating overΩ yield

minus intΩ11990611990111198911minus1199011119901 1205941 (V) ℎ (1199061 1199062 V) 119889119909

le minusintΩ119906119901+11 1198911minus1199011119901 1205941 (V) 119889119909 + 1205811 int

Ω11990611990111198911minus1199011119901 119889119909

(54)

By the monotonic property of exponential function andassumption (7) we have

0 lt 12058101 le 1205941 (V) exp(intV

V1205941 (119904) 119889119904) le 1205941 (V) 1198911119901 (V) (55)

It follows that (54) becomes

minus intΩ11990611990111198911minus1199011119901 1205941 (V) ℎ (1199061 1199062 V) 119889119909

le minus12058101 intΩ119906119901+11 119891minus1199011119901 119889119909 + 1205811 int

Ω11990611990111198911minus1199011119901 119889119909

(56)

Multiplying (56) by (119901 minus 1)(1 minus 120591((119901 minus 1)119901)) we get1198682 le 119901 minus 1

1 minus 120591 ((119901 minus 1) 119901) (minus12058101 intΩ 119906119901+11 119891minus1199011119901 119889119909+ 1205811 int

Ω11990611990111198911minus1199011119901 119889119909) le (119901 minus 1)

sdot (minus12058101 intΩ119906119901+11 119891minus1199011119901 119889119909 + 12058111 minus 120591 int

Ω11990611990111198911minus1199011119901 119889119909)

(57)

For the term 1198683 we easily achieve1198683 le 1199011205831 (int

Ω11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

le (119901 minus 1) 1205831 (intΩ11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

+ 1205831 intΩ11990611990111198911minus1199011119901 119889119909

(58)

Combining (44) (57) and (58) leads to

119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909 le (119901 minus 1)sdot (minus12058101 int

Ω119906119901+11 119891minus1199011119901 119889119909 + 12058111 minus 120591 int

Ω11990611990111198911minus1199011119901 119889119909)

+ (119901 minus 1) 1205831 (intΩ11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

+ 1205831 intΩ11990611990111198911minus1199011119901 119889119909

(59)

which implies that (51) holds The same arguments are usedto achieve (52) The proof ends

To establish the uniform boundedness of solutions weapply an iterative method based on the 119871119901-norm of solutionsinspired by the Moser-Alikakos type iterations (see [10 13])

8 Discrete Dynamics in Nature and Society

Lemma 8 For any 119905 lt 119879lowast then the solutions to (1) arebounded by

100381710038171003817100381711990611003817100381710038171003817119871infin(Ω) le 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

(60)

100381710038171003817100381711990621003817100381710038171003817119871infin(Ω) le 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(61)

Proof DenoteintΩ 11990611990111198911minus1199011119901 119889119909 byΨ(119905) Taking arbitrary positiveconstant 120588 gt 0 then it holds that

Ψ (119905) = int1198911119901le120588(12058101+1205831)1199061

11990611990111198911minus1199011119901 119889119909+ int1198911119901gt120588(12058101+1205831)1199061

11990611990111198911minus1199011119901 119889119909le 120588 (12058101 + 1205831) int

1198911119901le120588(12058101+1205831)1199061119906119901+11 119891minus1199011119901 119889119909

+ (120588 (12058101 + 1205831))1minus119901 int1198911119901gt120588(12058101+1205831)1199061

1199061119889119909le 120588 (12058101 + 1205831) int

Ω119906119901+11 119891minus1199011119901 119889119909

+ (120588 (12058101 + 1205831))1minus119901 intΩ1199061119889119909

(62)

Hence

minus (12058101 + 1205831) intΩ119906119901+11 119891minus1199011119901 119889119909

le minus1120588Ψ (119905) + 120588minus119901 (12058101 + 1205831)1minus119901 intΩ1199061119889119909

(63)

Substituting (63) into (51) we have

1119901 minus 1 119889119889119909Ψ (119905) le ( 12058111 minus 120591 + 119901119901 minus 11205831 minus 1120588)Ψ (119905)+ 120588minus119901 (12058101 + 1205831)1minus119901 int

Ω1199061119889119909

(64)

Take 1120588 gt 1205811(1 minus 120591) + (119901(119901 minus 1))1205831 Noticing Lemma 5and initial date 11990610 isin 1198621(Ω) we apply the Gronwall Lemmato obtain

Ψ (119905) le max120588minus119901 (12058101 + 1205831)1minus119901

sdot ( 1120588 minus 12058111 minus 120591 minus 119901119901 minus 11205831)minus1 intΩ1199061119889119909 Ψ (0)

(65)

taking limits leads to

lim119901rarrinfin

Ψ1119901 (119905) le max 120588minus1 (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω) (66)

Since

Ψ (119905) = intΩ11990611990111198911minus1199011119901 (V) 119889119909 ge 1198911minus1199011119901 (V) int

Ω1199061199011119889119909 (67)

combining (66) and (67) and taking limits as 120588minus1 rarr 1205811(1 minus120591) + 1205831 it is found that (60) holds The same processes areused to prove (61) which ends the proof of Lemma 8

Remark 9 In the above arguments we used well such factLetΩ be bounded and regular domain ofR119899 If 119906119871119901(Ω) le 119862where119862 is some positive constant and independent of119901 then119906119871119901(Ω) rarr 119906119871infin(Ω) as 119901 rarr infin

To end the arguments of uniform boundedness of solu-tions it follows to prove that 119879lowast = 119879max and 119879max = +infin bycontradiction

As in Lemmas 4ndash8 we define 1 2 and V for the uniquesolutions in (0 max) to

1119905 = Δ1 minus nabla sdot (11205941 (V) nablaV) + 120583111198921 (1 2) 119909 isin Ω 119905 gt 0

2119905 = Δ2 minus nabla sdot (21205942 (V) nablaV) + 120583211990621198922 (1 2) 119909 isin Ω 119905 gt 0

V119905 = 120591ΔV + ℎ (1 2 V) 119909 isin Ω 119905 gt 01205971120597] = 1205972120597] = 120597V120597] = 0 119909 isin 120597Ω 119905 gt 0

1 (119909 0) = 11990610 (119909) 2 (119909 0) = 11990620 (119909) V (119909 0) = V0 (119909)

119909 isin Ω

(68)

where max is defined in the same fashion as (32) and thetruncation ℎ(1199061 1199062 V) is defined as follows

ℎ (1199061 1199062 V) fl ℎ (1199061 1199062 V) if V le Vℎ (1199061 1199062 V) if V gt V (69)

Then it is easy to see that100381710038171003817100381711003817100381710038171003817119871infin(Ω) le 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

100381710038171003817100381721003817100381710038171003817119871infin(Ω) le 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(70)

as far as V le VBy analogy with 119879lowast we define

lowast fl max if V le V in (0 max) V if V119871infin(Ωmax )

gt V (71)

Discrete Dynamics in Nature and Society 9

where V lt max satisfies that V119871infin(ΩV ) le V and

V119871infin(ΩV+120575) gt V for any 120575 gt 0 To prove lowast = max wesuppose

lowast lt max (72)

then ℎ(1 2 V) le ℎ(1199061 1199062 V) = ℎ(1199061 1199062 V) lt 0 It follows toeasily check that V is a strict upper-solution to

V119905 minus 120591ΔV = ℎ (1 2 V) in Ω times (0 lowast) 120597V120597] = 0 on 120597Ω times (0 lowast)

V (119909 0) = V0 (119909) in Ω(73)

This implies V(119909 119905) lt V forall119909 isin Ω 119905 isin (0 lowast) whichcontradicts (72) and proves lowast = max as well as max = infin

Thanks to V le V we show that (1 2 V) is also a solutionto (6) So we have

119879lowast = 119879max ge max = infin (74)

The proof of Theorem 3 is completed

Competing Interests

The author declares that they have no competing interests

Acknowledgments

Thiswork is supported by the ChinaNational Natural ScienceFoundation (nos 11361055 and 11261053) and the NaturalScience ResearchKey Project of University of Anhui Province(no KJ2015A251)

References

[1] J D Murray Mathematical Biology vol 19 Springer BerlinGermany 2nd edition 1993

[2] M Winkler ldquoBoundedness in the higher-dimensionalparabolic-parabolic chemotaxis system with logistic sourcerdquoCommunications in Partial Differential Equations vol 35 no 8pp 1516ndash1537 2010

[3] T Hillen and K J Painter ldquoA userrsquos guide to PDE models forchemotaxisrdquo Journal of Mathematical Biology vol 58 no 1-2pp 183ndash217 2009

[4] E F Keller and L A Segel ldquoInitiation of slimemold aggregationviewed as an instabilityrdquo Journal of Theoretical Biology vol 26no 3 pp 399ndash415 1970

[5] M Winkler ldquoAbsence of collapse in a parabolic chemo-taxis system with signal-dependent sensitivityrdquo MathematischeNachrichten vol 283 no 11 pp 1664ndash1673 2010

[6] J I Tello and M Winkler ldquoStabilization in a two-specieschemotaxis system with a logistic sourcerdquo Nonlinearity vol 25no 5 pp 1413ndash1425 2012

[7] M Negreanu and J I Tello ldquoAsymptotic stability of a twospecies chemotaxis systemwith non-diffusive chemoattractantrdquoJournal of Differential Equations vol 258 no 5 pp 1592ndash16172015

[8] Q Zhang and Y Li ldquoGlobal boundedness of solutions toa two-species chemotaxis systemrdquo Zeitschrift fur angewandteMathematik und Physik vol 66 no 1 pp 83ndash93 2015

[9] C Mu L Wang P Zheng and Q Zhang ldquoGlobal existenceand boundedness of classical solutions to a parabolic-parabolicchemotaxis systemrdquo Nonlinear Analysis Real World Applica-tions vol 14 no 3 pp 1634ndash1642 2013

[10] Y Tao ldquoBoundedness in a chemotaxis model with oxygenconsumption by bacteriardquo Journal ofMathematical Analysis andApplications vol 381 no 2 pp 521ndash529 2011

[11] D Horstmann and M Winkler ldquoBoundedness vs blow-up in achemotaxis systemrdquo Journal of Differential Equations vol 215no 1 pp 52ndash107 2005

[12] M Winkler ldquoAggregation vs global diffusive behavior in thehigher-dimensional Keller-Segel modelrdquo Journal of DifferentialEquations vol 248 no 12 pp 2889ndash2905 2010

[13] PQuittner andP Souplet Superlinear Parabolic Problems Blow-Up Global Existence and Steady States Birkhauser AdvancedTexts Birkhauser Basel Switzerland 2007

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Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

Lemma 6 Letting 119901 gt 1 and under assumption (11) then thefollowing estimates hold

119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909le 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990111205941 (V) 1198911minus1199011119901 ℎ (1199061 1199062 V) 119889119909+ 1199011205831 int

Ω11990611990111198911minus1199011119901 1198921 (1199061 1199062) 119889119909

(40)

119889119889119905 intΩ 11990611990121198911minus1199012119901 119889119909le 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990121205942 (V) 1198911minus1199012119901 ℎ (1199061 1199062 V) 119889119909+ 1199011205832 int

Ω11990611990121198911minus1199012119901 1198922 (1199061 1199062) 119889119909

(41)

where 119891119894119901(V) is defined by

119891119894119901 (V) = exp( 11 minus 120591 ((119901 minus 1) 119901) intV

V120594119894 (119904) 119889119904)

119894 = 1 2(42)

Proof It is easy to check

11989110158401119901 (V) = 11 minus 120591 ((119901 minus 1) 119901)1205941 (V) 1198911119901 (V) 11989110158402119901 (V) = 11 minus 120591 ((119901 minus 1) 119901)1205942 (V) 1198912119901 (V)

(43)

Then for any 119901 gt 1 we have119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909 = 119901int

Ω119906119901minus11 11990611199051198911minus1199011119901 119889119909 + (1 minus 119901)

sdot intΩ1199061199011119891minus1199011119901 11989110158401119901V119905119889119909 = 119901int

Ω119906119901minus11 1198911minus1199011119901 [Δ1199061 minus nabla

sdot (11990611205941 (V) nablaV) + 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1)11990611205941 (V) ΔV]119889119909

+ 1 minus 1199011 minus 120591 ((119901 minus 1) 119901) intΩ 11990611990111198911minus1199011119901 1205941 (V)

sdot ℎ (1199061 1199062 V) 119889119909+ 1199011205831 int

Ω11990611990111198911minus1199011119901 1198921 (1199061 1199062) 119889119909 fl 1198681 + 1198682 + 1198683

(44)

Clearly if 1198681 le 0 then (40) is a consequence of (44) We inthe following verify that the result holds

In fact

1198681 = 119901intΩ119906119901minus11 1198911minus1199011119901 [Δ1199061 minus nabla sdot (11990611205941 (V) nablaV)

+ 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1) (nabla sdot (11990611205941 (V) nablaV) minus nabla (11990611205941 (V))

sdot nablaV)] 119889119909 = 119901intΩ119906119901minus11 1198911minus1199011119901 [nabla

sdot (1198911119901nabla (1199061119891minus11119901 )) minus 120591 (1 minus 119901)119901 minus 120591 (119901 minus 1) (1205941 (V) nabla1199061 sdot nablaV

+ 119906112059410158401 (V) |nablaV|2)] 119889119909 = 119901 (1 minus 119901)

sdot intΩ119906119901minus21 1198912minus1199011119901 [1198911119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162

minus 1205911199061 (1205941 (V) nabla1199061 sdot nablaV + 119906112059410158401 (V) |nablaV|2)1198911119901 (119901 minus 120591 (119901 minus 1)) ] 119889119909= 119901 (1 minus 119901)int

Ω119906119901minus21 1198913minus1199011119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162 119889119909

+ 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ 119906119901minus11 1198911minus1199011119901 1205941 (V) nabla1199061 sdot nablaV 119889119909

+ 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ 11990611990111198911minus1199011119901 12059410158401 (V) |nablaV|2 119889119909 fl 1198691

+ 1198692 + 1198693

(45)

Since nabla1199061 = nabla(1198911119901(1199061119891minus11119901 )) multiplying both sides by1205941(V)nablaV we get1205941 (V) nabla1199061 sdot nablaV

= 11989111199011205941 (V) nabla (1199061119891minus11119901 ) sdot nablaV+ 11 minus 120591 ((119901 minus 1) 119901)119906112059421 (V) |nablaV|2

(46)

and by 120576-Youngrsquos inequality1205941 (V) nabla (1199061119891minus11119901 ) sdot nablaV

le 119901 minus 120591 (119901 minus 1)120591

1198911119901119906110038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162

+ 120591119901 minus 120591 (119901 minus 1) 1199061119891111990112059421 (V) |nablaV|2

(47)

Combining (46) and (47) and substituting it into 1198692 we derive1198692 le 119901 (119901 minus 1)int

Ω119906119901minus21 1198913minus1199011119901 10038161003816100381610038161003816nabla (1199061119891minus11119901 )100381610038161003816100381610038162 119889119909

+ 120591119901 (119901 minus 1) (120591 + 119901)(119901 minus 120591 (119901 minus 1))2 int

Ω11990611990111198911minus1199011119901 12059421 (V) |nablaV|2 119889119909

(48)

Discrete Dynamics in Nature and Society 7

Inserting (48) into (45) yields

1198681 le 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ (

120591 + 119901(119901 minus 120591 (119901 minus 1))12059421 (V)

+ 12059410158401 (V)) 11990611990111198911minus1199011119901 |nablaV|2 119889119909 = 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1)

sdot intΩ[ 1(1 minus 120591) (1 minus 1205912120591 + (1 minus 120591) 119901)12059421 (V)

+ 12059410158401 (V)] 11990611990111198911minus1199011119901 |nablaV|2 119889119909 le 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1)

sdot intΩ( 1(1 minus 120591)12059421 (V) + 12059410158401 (V)) 11990611990111198911minus1199011119901 |nablaV|2 119889119909

(49)

we deduce from assumption (11) that 1198681 le 0 for any 119901 gt 1and the proof ends In the same way we achieve (41) Thiscompletes the proof of Lemma 6

In view of V isin 119862((0 119879max) 1198621(Ω)) and initial date V ltV0 lt V we consider 119879lowast gt 0 defined by

119879lowast fl 119879max if V le V in (0 119879max) 119879V if V119871infin(Ω119879max )

gt V (50)

where 119879V lt 119879max satisfies that V119871infin(Ω119879V ) le V andV119871infin(Ω119879V+120575) gt V for any 120575 gt 0Lemma7 Letting119901 gt 1 and for any 119905 lt 119879lowast then the solutionsto (1) satisfy

1119901 minus 1 119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909le minus (12058101 + 1205831) int

Ω119906119901+11 119891minus1199011119901 119889119909

+ ( 12058111 minus 120591 + 119901119901 minus 11205831)intΩ11990611990111198911minus1199011119901 119889119909

(51)

1119901 minus 1 119889119889119905 intΩ 11990611990121198911minus1199012119901 119889119909le minus (12058102 + 1205832) int

Ω119906119901+12 119891minus1199012119901 119889119909

+ ( 12058121 minus 120591 + 119901119901 minus 11205832)intΩ11990611990121198911minus1199012119901 119889119909

(52)

Proof In order to prove Lemma 7 we consider the terms 1198682and 1198683 appearing in the proof of Lemma 6

By the nonnegativity for 1199061 1199062 and assumption (6) weobtain

minusℎ (1199061 1199062 V) le minus1199061 + 12058111205941 (V) (53)

Multiplying (53) by 11990611990111198911minus1199011119901 1205941(V) and integrating overΩ yield

minus intΩ11990611990111198911minus1199011119901 1205941 (V) ℎ (1199061 1199062 V) 119889119909

le minusintΩ119906119901+11 1198911minus1199011119901 1205941 (V) 119889119909 + 1205811 int

Ω11990611990111198911minus1199011119901 119889119909

(54)

By the monotonic property of exponential function andassumption (7) we have

0 lt 12058101 le 1205941 (V) exp(intV

V1205941 (119904) 119889119904) le 1205941 (V) 1198911119901 (V) (55)

It follows that (54) becomes

minus intΩ11990611990111198911minus1199011119901 1205941 (V) ℎ (1199061 1199062 V) 119889119909

le minus12058101 intΩ119906119901+11 119891minus1199011119901 119889119909 + 1205811 int

Ω11990611990111198911minus1199011119901 119889119909

(56)

Multiplying (56) by (119901 minus 1)(1 minus 120591((119901 minus 1)119901)) we get1198682 le 119901 minus 1

1 minus 120591 ((119901 minus 1) 119901) (minus12058101 intΩ 119906119901+11 119891minus1199011119901 119889119909+ 1205811 int

Ω11990611990111198911minus1199011119901 119889119909) le (119901 minus 1)

sdot (minus12058101 intΩ119906119901+11 119891minus1199011119901 119889119909 + 12058111 minus 120591 int

Ω11990611990111198911minus1199011119901 119889119909)

(57)

For the term 1198683 we easily achieve1198683 le 1199011205831 (int

Ω11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

le (119901 minus 1) 1205831 (intΩ11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

+ 1205831 intΩ11990611990111198911minus1199011119901 119889119909

(58)

Combining (44) (57) and (58) leads to

119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909 le (119901 minus 1)sdot (minus12058101 int

Ω119906119901+11 119891minus1199011119901 119889119909 + 12058111 minus 120591 int

Ω11990611990111198911minus1199011119901 119889119909)

+ (119901 minus 1) 1205831 (intΩ11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

+ 1205831 intΩ11990611990111198911minus1199011119901 119889119909

(59)

which implies that (51) holds The same arguments are usedto achieve (52) The proof ends

To establish the uniform boundedness of solutions weapply an iterative method based on the 119871119901-norm of solutionsinspired by the Moser-Alikakos type iterations (see [10 13])

8 Discrete Dynamics in Nature and Society

Lemma 8 For any 119905 lt 119879lowast then the solutions to (1) arebounded by

100381710038171003817100381711990611003817100381710038171003817119871infin(Ω) le 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

(60)

100381710038171003817100381711990621003817100381710038171003817119871infin(Ω) le 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(61)

Proof DenoteintΩ 11990611990111198911minus1199011119901 119889119909 byΨ(119905) Taking arbitrary positiveconstant 120588 gt 0 then it holds that

Ψ (119905) = int1198911119901le120588(12058101+1205831)1199061

11990611990111198911minus1199011119901 119889119909+ int1198911119901gt120588(12058101+1205831)1199061

11990611990111198911minus1199011119901 119889119909le 120588 (12058101 + 1205831) int

1198911119901le120588(12058101+1205831)1199061119906119901+11 119891minus1199011119901 119889119909

+ (120588 (12058101 + 1205831))1minus119901 int1198911119901gt120588(12058101+1205831)1199061

1199061119889119909le 120588 (12058101 + 1205831) int

Ω119906119901+11 119891minus1199011119901 119889119909

+ (120588 (12058101 + 1205831))1minus119901 intΩ1199061119889119909

(62)

Hence

minus (12058101 + 1205831) intΩ119906119901+11 119891minus1199011119901 119889119909

le minus1120588Ψ (119905) + 120588minus119901 (12058101 + 1205831)1minus119901 intΩ1199061119889119909

(63)

Substituting (63) into (51) we have

1119901 minus 1 119889119889119909Ψ (119905) le ( 12058111 minus 120591 + 119901119901 minus 11205831 minus 1120588)Ψ (119905)+ 120588minus119901 (12058101 + 1205831)1minus119901 int

Ω1199061119889119909

(64)

Take 1120588 gt 1205811(1 minus 120591) + (119901(119901 minus 1))1205831 Noticing Lemma 5and initial date 11990610 isin 1198621(Ω) we apply the Gronwall Lemmato obtain

Ψ (119905) le max120588minus119901 (12058101 + 1205831)1minus119901

sdot ( 1120588 minus 12058111 minus 120591 minus 119901119901 minus 11205831)minus1 intΩ1199061119889119909 Ψ (0)

(65)

taking limits leads to

lim119901rarrinfin

Ψ1119901 (119905) le max 120588minus1 (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω) (66)

Since

Ψ (119905) = intΩ11990611990111198911minus1199011119901 (V) 119889119909 ge 1198911minus1199011119901 (V) int

Ω1199061199011119889119909 (67)

combining (66) and (67) and taking limits as 120588minus1 rarr 1205811(1 minus120591) + 1205831 it is found that (60) holds The same processes areused to prove (61) which ends the proof of Lemma 8

Remark 9 In the above arguments we used well such factLetΩ be bounded and regular domain ofR119899 If 119906119871119901(Ω) le 119862where119862 is some positive constant and independent of119901 then119906119871119901(Ω) rarr 119906119871infin(Ω) as 119901 rarr infin

To end the arguments of uniform boundedness of solu-tions it follows to prove that 119879lowast = 119879max and 119879max = +infin bycontradiction

As in Lemmas 4ndash8 we define 1 2 and V for the uniquesolutions in (0 max) to

1119905 = Δ1 minus nabla sdot (11205941 (V) nablaV) + 120583111198921 (1 2) 119909 isin Ω 119905 gt 0

2119905 = Δ2 minus nabla sdot (21205942 (V) nablaV) + 120583211990621198922 (1 2) 119909 isin Ω 119905 gt 0

V119905 = 120591ΔV + ℎ (1 2 V) 119909 isin Ω 119905 gt 01205971120597] = 1205972120597] = 120597V120597] = 0 119909 isin 120597Ω 119905 gt 0

1 (119909 0) = 11990610 (119909) 2 (119909 0) = 11990620 (119909) V (119909 0) = V0 (119909)

119909 isin Ω

(68)

where max is defined in the same fashion as (32) and thetruncation ℎ(1199061 1199062 V) is defined as follows

ℎ (1199061 1199062 V) fl ℎ (1199061 1199062 V) if V le Vℎ (1199061 1199062 V) if V gt V (69)

Then it is easy to see that100381710038171003817100381711003817100381710038171003817119871infin(Ω) le 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

100381710038171003817100381721003817100381710038171003817119871infin(Ω) le 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(70)

as far as V le VBy analogy with 119879lowast we define

lowast fl max if V le V in (0 max) V if V119871infin(Ωmax )

gt V (71)

Discrete Dynamics in Nature and Society 9

where V lt max satisfies that V119871infin(ΩV ) le V and

V119871infin(ΩV+120575) gt V for any 120575 gt 0 To prove lowast = max wesuppose

lowast lt max (72)

then ℎ(1 2 V) le ℎ(1199061 1199062 V) = ℎ(1199061 1199062 V) lt 0 It follows toeasily check that V is a strict upper-solution to

V119905 minus 120591ΔV = ℎ (1 2 V) in Ω times (0 lowast) 120597V120597] = 0 on 120597Ω times (0 lowast)

V (119909 0) = V0 (119909) in Ω(73)

This implies V(119909 119905) lt V forall119909 isin Ω 119905 isin (0 lowast) whichcontradicts (72) and proves lowast = max as well as max = infin

Thanks to V le V we show that (1 2 V) is also a solutionto (6) So we have

119879lowast = 119879max ge max = infin (74)

The proof of Theorem 3 is completed

Competing Interests

The author declares that they have no competing interests

Acknowledgments

Thiswork is supported by the ChinaNational Natural ScienceFoundation (nos 11361055 and 11261053) and the NaturalScience ResearchKey Project of University of Anhui Province(no KJ2015A251)

References

[1] J D Murray Mathematical Biology vol 19 Springer BerlinGermany 2nd edition 1993

[2] M Winkler ldquoBoundedness in the higher-dimensionalparabolic-parabolic chemotaxis system with logistic sourcerdquoCommunications in Partial Differential Equations vol 35 no 8pp 1516ndash1537 2010

[3] T Hillen and K J Painter ldquoA userrsquos guide to PDE models forchemotaxisrdquo Journal of Mathematical Biology vol 58 no 1-2pp 183ndash217 2009

[4] E F Keller and L A Segel ldquoInitiation of slimemold aggregationviewed as an instabilityrdquo Journal of Theoretical Biology vol 26no 3 pp 399ndash415 1970

[5] M Winkler ldquoAbsence of collapse in a parabolic chemo-taxis system with signal-dependent sensitivityrdquo MathematischeNachrichten vol 283 no 11 pp 1664ndash1673 2010

[6] J I Tello and M Winkler ldquoStabilization in a two-specieschemotaxis system with a logistic sourcerdquo Nonlinearity vol 25no 5 pp 1413ndash1425 2012

[7] M Negreanu and J I Tello ldquoAsymptotic stability of a twospecies chemotaxis systemwith non-diffusive chemoattractantrdquoJournal of Differential Equations vol 258 no 5 pp 1592ndash16172015

[8] Q Zhang and Y Li ldquoGlobal boundedness of solutions toa two-species chemotaxis systemrdquo Zeitschrift fur angewandteMathematik und Physik vol 66 no 1 pp 83ndash93 2015

[9] C Mu L Wang P Zheng and Q Zhang ldquoGlobal existenceand boundedness of classical solutions to a parabolic-parabolicchemotaxis systemrdquo Nonlinear Analysis Real World Applica-tions vol 14 no 3 pp 1634ndash1642 2013

[10] Y Tao ldquoBoundedness in a chemotaxis model with oxygenconsumption by bacteriardquo Journal ofMathematical Analysis andApplications vol 381 no 2 pp 521ndash529 2011

[11] D Horstmann and M Winkler ldquoBoundedness vs blow-up in achemotaxis systemrdquo Journal of Differential Equations vol 215no 1 pp 52ndash107 2005

[12] M Winkler ldquoAggregation vs global diffusive behavior in thehigher-dimensional Keller-Segel modelrdquo Journal of DifferentialEquations vol 248 no 12 pp 2889ndash2905 2010

[13] PQuittner andP Souplet Superlinear Parabolic Problems Blow-Up Global Existence and Steady States Birkhauser AdvancedTexts Birkhauser Basel Switzerland 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

Inserting (48) into (45) yields

1198681 le 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1) intΩ (

120591 + 119901(119901 minus 120591 (119901 minus 1))12059421 (V)

+ 12059410158401 (V)) 11990611990111198911minus1199011119901 |nablaV|2 119889119909 = 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1)

sdot intΩ[ 1(1 minus 120591) (1 minus 1205912120591 + (1 minus 120591) 119901)12059421 (V)

+ 12059410158401 (V)] 11990611990111198911minus1199011119901 |nablaV|2 119889119909 le 120591119901 (119901 minus 1)119901 minus 120591 (119901 minus 1)

sdot intΩ( 1(1 minus 120591)12059421 (V) + 12059410158401 (V)) 11990611990111198911minus1199011119901 |nablaV|2 119889119909

(49)

we deduce from assumption (11) that 1198681 le 0 for any 119901 gt 1and the proof ends In the same way we achieve (41) Thiscompletes the proof of Lemma 6

In view of V isin 119862((0 119879max) 1198621(Ω)) and initial date V ltV0 lt V we consider 119879lowast gt 0 defined by

119879lowast fl 119879max if V le V in (0 119879max) 119879V if V119871infin(Ω119879max )

gt V (50)

where 119879V lt 119879max satisfies that V119871infin(Ω119879V ) le V andV119871infin(Ω119879V+120575) gt V for any 120575 gt 0Lemma7 Letting119901 gt 1 and for any 119905 lt 119879lowast then the solutionsto (1) satisfy

1119901 minus 1 119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909le minus (12058101 + 1205831) int

Ω119906119901+11 119891minus1199011119901 119889119909

+ ( 12058111 minus 120591 + 119901119901 minus 11205831)intΩ11990611990111198911minus1199011119901 119889119909

(51)

1119901 minus 1 119889119889119905 intΩ 11990611990121198911minus1199012119901 119889119909le minus (12058102 + 1205832) int

Ω119906119901+12 119891minus1199012119901 119889119909

+ ( 12058121 minus 120591 + 119901119901 minus 11205832)intΩ11990611990121198911minus1199012119901 119889119909

(52)

Proof In order to prove Lemma 7 we consider the terms 1198682and 1198683 appearing in the proof of Lemma 6

By the nonnegativity for 1199061 1199062 and assumption (6) weobtain

minusℎ (1199061 1199062 V) le minus1199061 + 12058111205941 (V) (53)

Multiplying (53) by 11990611990111198911minus1199011119901 1205941(V) and integrating overΩ yield

minus intΩ11990611990111198911minus1199011119901 1205941 (V) ℎ (1199061 1199062 V) 119889119909

le minusintΩ119906119901+11 1198911minus1199011119901 1205941 (V) 119889119909 + 1205811 int

Ω11990611990111198911minus1199011119901 119889119909

(54)

By the monotonic property of exponential function andassumption (7) we have

0 lt 12058101 le 1205941 (V) exp(intV

V1205941 (119904) 119889119904) le 1205941 (V) 1198911119901 (V) (55)

It follows that (54) becomes

minus intΩ11990611990111198911minus1199011119901 1205941 (V) ℎ (1199061 1199062 V) 119889119909

le minus12058101 intΩ119906119901+11 119891minus1199011119901 119889119909 + 1205811 int

Ω11990611990111198911minus1199011119901 119889119909

(56)

Multiplying (56) by (119901 minus 1)(1 minus 120591((119901 minus 1)119901)) we get1198682 le 119901 minus 1

1 minus 120591 ((119901 minus 1) 119901) (minus12058101 intΩ 119906119901+11 119891minus1199011119901 119889119909+ 1205811 int

Ω11990611990111198911minus1199011119901 119889119909) le (119901 minus 1)

sdot (minus12058101 intΩ119906119901+11 119891minus1199011119901 119889119909 + 12058111 minus 120591 int

Ω11990611990111198911minus1199011119901 119889119909)

(57)

For the term 1198683 we easily achieve1198683 le 1199011205831 (int

Ω11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

le (119901 minus 1) 1205831 (intΩ11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

+ 1205831 intΩ11990611990111198911minus1199011119901 119889119909

(58)

Combining (44) (57) and (58) leads to

119889119889119905 intΩ 11990611990111198911minus1199011119901 119889119909 le (119901 minus 1)sdot (minus12058101 int

Ω119906119901+11 119891minus1199011119901 119889119909 + 12058111 minus 120591 int

Ω11990611990111198911minus1199011119901 119889119909)

+ (119901 minus 1) 1205831 (intΩ11990611990111198911minus1199011119901 119889119909 minus int

Ω119906119901+11 119891minus1199011119901 119889119909)

+ 1205831 intΩ11990611990111198911minus1199011119901 119889119909

(59)

which implies that (51) holds The same arguments are usedto achieve (52) The proof ends

To establish the uniform boundedness of solutions weapply an iterative method based on the 119871119901-norm of solutionsinspired by the Moser-Alikakos type iterations (see [10 13])

8 Discrete Dynamics in Nature and Society

Lemma 8 For any 119905 lt 119879lowast then the solutions to (1) arebounded by

100381710038171003817100381711990611003817100381710038171003817119871infin(Ω) le 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

(60)

100381710038171003817100381711990621003817100381710038171003817119871infin(Ω) le 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(61)

Proof DenoteintΩ 11990611990111198911minus1199011119901 119889119909 byΨ(119905) Taking arbitrary positiveconstant 120588 gt 0 then it holds that

Ψ (119905) = int1198911119901le120588(12058101+1205831)1199061

11990611990111198911minus1199011119901 119889119909+ int1198911119901gt120588(12058101+1205831)1199061

11990611990111198911minus1199011119901 119889119909le 120588 (12058101 + 1205831) int

1198911119901le120588(12058101+1205831)1199061119906119901+11 119891minus1199011119901 119889119909

+ (120588 (12058101 + 1205831))1minus119901 int1198911119901gt120588(12058101+1205831)1199061

1199061119889119909le 120588 (12058101 + 1205831) int

Ω119906119901+11 119891minus1199011119901 119889119909

+ (120588 (12058101 + 1205831))1minus119901 intΩ1199061119889119909

(62)

Hence

minus (12058101 + 1205831) intΩ119906119901+11 119891minus1199011119901 119889119909

le minus1120588Ψ (119905) + 120588minus119901 (12058101 + 1205831)1minus119901 intΩ1199061119889119909

(63)

Substituting (63) into (51) we have

1119901 minus 1 119889119889119909Ψ (119905) le ( 12058111 minus 120591 + 119901119901 minus 11205831 minus 1120588)Ψ (119905)+ 120588minus119901 (12058101 + 1205831)1minus119901 int

Ω1199061119889119909

(64)

Take 1120588 gt 1205811(1 minus 120591) + (119901(119901 minus 1))1205831 Noticing Lemma 5and initial date 11990610 isin 1198621(Ω) we apply the Gronwall Lemmato obtain

Ψ (119905) le max120588minus119901 (12058101 + 1205831)1minus119901

sdot ( 1120588 minus 12058111 minus 120591 minus 119901119901 minus 11205831)minus1 intΩ1199061119889119909 Ψ (0)

(65)

taking limits leads to

lim119901rarrinfin

Ψ1119901 (119905) le max 120588minus1 (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω) (66)

Since

Ψ (119905) = intΩ11990611990111198911minus1199011119901 (V) 119889119909 ge 1198911minus1199011119901 (V) int

Ω1199061199011119889119909 (67)

combining (66) and (67) and taking limits as 120588minus1 rarr 1205811(1 minus120591) + 1205831 it is found that (60) holds The same processes areused to prove (61) which ends the proof of Lemma 8

Remark 9 In the above arguments we used well such factLetΩ be bounded and regular domain ofR119899 If 119906119871119901(Ω) le 119862where119862 is some positive constant and independent of119901 then119906119871119901(Ω) rarr 119906119871infin(Ω) as 119901 rarr infin

To end the arguments of uniform boundedness of solu-tions it follows to prove that 119879lowast = 119879max and 119879max = +infin bycontradiction

As in Lemmas 4ndash8 we define 1 2 and V for the uniquesolutions in (0 max) to

1119905 = Δ1 minus nabla sdot (11205941 (V) nablaV) + 120583111198921 (1 2) 119909 isin Ω 119905 gt 0

2119905 = Δ2 minus nabla sdot (21205942 (V) nablaV) + 120583211990621198922 (1 2) 119909 isin Ω 119905 gt 0

V119905 = 120591ΔV + ℎ (1 2 V) 119909 isin Ω 119905 gt 01205971120597] = 1205972120597] = 120597V120597] = 0 119909 isin 120597Ω 119905 gt 0

1 (119909 0) = 11990610 (119909) 2 (119909 0) = 11990620 (119909) V (119909 0) = V0 (119909)

119909 isin Ω

(68)

where max is defined in the same fashion as (32) and thetruncation ℎ(1199061 1199062 V) is defined as follows

ℎ (1199061 1199062 V) fl ℎ (1199061 1199062 V) if V le Vℎ (1199061 1199062 V) if V gt V (69)

Then it is easy to see that100381710038171003817100381711003817100381710038171003817119871infin(Ω) le 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

100381710038171003817100381721003817100381710038171003817119871infin(Ω) le 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(70)

as far as V le VBy analogy with 119879lowast we define

lowast fl max if V le V in (0 max) V if V119871infin(Ωmax )

gt V (71)

Discrete Dynamics in Nature and Society 9

where V lt max satisfies that V119871infin(ΩV ) le V and

V119871infin(ΩV+120575) gt V for any 120575 gt 0 To prove lowast = max wesuppose

lowast lt max (72)

then ℎ(1 2 V) le ℎ(1199061 1199062 V) = ℎ(1199061 1199062 V) lt 0 It follows toeasily check that V is a strict upper-solution to

V119905 minus 120591ΔV = ℎ (1 2 V) in Ω times (0 lowast) 120597V120597] = 0 on 120597Ω times (0 lowast)

V (119909 0) = V0 (119909) in Ω(73)

This implies V(119909 119905) lt V forall119909 isin Ω 119905 isin (0 lowast) whichcontradicts (72) and proves lowast = max as well as max = infin

Thanks to V le V we show that (1 2 V) is also a solutionto (6) So we have

119879lowast = 119879max ge max = infin (74)

The proof of Theorem 3 is completed

Competing Interests

The author declares that they have no competing interests

Acknowledgments

Thiswork is supported by the ChinaNational Natural ScienceFoundation (nos 11361055 and 11261053) and the NaturalScience ResearchKey Project of University of Anhui Province(no KJ2015A251)

References

[1] J D Murray Mathematical Biology vol 19 Springer BerlinGermany 2nd edition 1993

[2] M Winkler ldquoBoundedness in the higher-dimensionalparabolic-parabolic chemotaxis system with logistic sourcerdquoCommunications in Partial Differential Equations vol 35 no 8pp 1516ndash1537 2010

[3] T Hillen and K J Painter ldquoA userrsquos guide to PDE models forchemotaxisrdquo Journal of Mathematical Biology vol 58 no 1-2pp 183ndash217 2009

[4] E F Keller and L A Segel ldquoInitiation of slimemold aggregationviewed as an instabilityrdquo Journal of Theoretical Biology vol 26no 3 pp 399ndash415 1970

[5] M Winkler ldquoAbsence of collapse in a parabolic chemo-taxis system with signal-dependent sensitivityrdquo MathematischeNachrichten vol 283 no 11 pp 1664ndash1673 2010

[6] J I Tello and M Winkler ldquoStabilization in a two-specieschemotaxis system with a logistic sourcerdquo Nonlinearity vol 25no 5 pp 1413ndash1425 2012

[7] M Negreanu and J I Tello ldquoAsymptotic stability of a twospecies chemotaxis systemwith non-diffusive chemoattractantrdquoJournal of Differential Equations vol 258 no 5 pp 1592ndash16172015

[8] Q Zhang and Y Li ldquoGlobal boundedness of solutions toa two-species chemotaxis systemrdquo Zeitschrift fur angewandteMathematik und Physik vol 66 no 1 pp 83ndash93 2015

[9] C Mu L Wang P Zheng and Q Zhang ldquoGlobal existenceand boundedness of classical solutions to a parabolic-parabolicchemotaxis systemrdquo Nonlinear Analysis Real World Applica-tions vol 14 no 3 pp 1634ndash1642 2013

[10] Y Tao ldquoBoundedness in a chemotaxis model with oxygenconsumption by bacteriardquo Journal ofMathematical Analysis andApplications vol 381 no 2 pp 521ndash529 2011

[11] D Horstmann and M Winkler ldquoBoundedness vs blow-up in achemotaxis systemrdquo Journal of Differential Equations vol 215no 1 pp 52ndash107 2005

[12] M Winkler ldquoAggregation vs global diffusive behavior in thehigher-dimensional Keller-Segel modelrdquo Journal of DifferentialEquations vol 248 no 12 pp 2889ndash2905 2010

[13] PQuittner andP Souplet Superlinear Parabolic Problems Blow-Up Global Existence and Steady States Birkhauser AdvancedTexts Birkhauser Basel Switzerland 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

Lemma 8 For any 119905 lt 119879lowast then the solutions to (1) arebounded by

100381710038171003817100381711990611003817100381710038171003817119871infin(Ω) le 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

(60)

100381710038171003817100381711990621003817100381710038171003817119871infin(Ω) le 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(61)

Proof DenoteintΩ 11990611990111198911minus1199011119901 119889119909 byΨ(119905) Taking arbitrary positiveconstant 120588 gt 0 then it holds that

Ψ (119905) = int1198911119901le120588(12058101+1205831)1199061

11990611990111198911minus1199011119901 119889119909+ int1198911119901gt120588(12058101+1205831)1199061

11990611990111198911minus1199011119901 119889119909le 120588 (12058101 + 1205831) int

1198911119901le120588(12058101+1205831)1199061119906119901+11 119891minus1199011119901 119889119909

+ (120588 (12058101 + 1205831))1minus119901 int1198911119901gt120588(12058101+1205831)1199061

1199061119889119909le 120588 (12058101 + 1205831) int

Ω119906119901+11 119891minus1199011119901 119889119909

+ (120588 (12058101 + 1205831))1minus119901 intΩ1199061119889119909

(62)

Hence

minus (12058101 + 1205831) intΩ119906119901+11 119891minus1199011119901 119889119909

le minus1120588Ψ (119905) + 120588minus119901 (12058101 + 1205831)1minus119901 intΩ1199061119889119909

(63)

Substituting (63) into (51) we have

1119901 minus 1 119889119889119909Ψ (119905) le ( 12058111 minus 120591 + 119901119901 minus 11205831 minus 1120588)Ψ (119905)+ 120588minus119901 (12058101 + 1205831)1minus119901 int

Ω1199061119889119909

(64)

Take 1120588 gt 1205811(1 minus 120591) + (119901(119901 minus 1))1205831 Noticing Lemma 5and initial date 11990610 isin 1198621(Ω) we apply the Gronwall Lemmato obtain

Ψ (119905) le max120588minus119901 (12058101 + 1205831)1minus119901

sdot ( 1120588 minus 12058111 minus 120591 minus 119901119901 minus 11205831)minus1 intΩ1199061119889119909 Ψ (0)

(65)

taking limits leads to

lim119901rarrinfin

Ψ1119901 (119905) le max 120588minus1 (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω) (66)

Since

Ψ (119905) = intΩ11990611990111198911minus1199011119901 (V) 119889119909 ge 1198911minus1199011119901 (V) int

Ω1199061199011119889119909 (67)

combining (66) and (67) and taking limits as 120588minus1 rarr 1205811(1 minus120591) + 1205831 it is found that (60) holds The same processes areused to prove (61) which ends the proof of Lemma 8

Remark 9 In the above arguments we used well such factLetΩ be bounded and regular domain ofR119899 If 119906119871119901(Ω) le 119862where119862 is some positive constant and independent of119901 then119906119871119901(Ω) rarr 119906119871infin(Ω) as 119901 rarr infin

To end the arguments of uniform boundedness of solu-tions it follows to prove that 119879lowast = 119879max and 119879max = +infin bycontradiction

As in Lemmas 4ndash8 we define 1 2 and V for the uniquesolutions in (0 max) to

1119905 = Δ1 minus nabla sdot (11205941 (V) nablaV) + 120583111198921 (1 2) 119909 isin Ω 119905 gt 0

2119905 = Δ2 minus nabla sdot (21205942 (V) nablaV) + 120583211990621198922 (1 2) 119909 isin Ω 119905 gt 0

V119905 = 120591ΔV + ℎ (1 2 V) 119909 isin Ω 119905 gt 01205971120597] = 1205972120597] = 120597V120597] = 0 119909 isin 120597Ω 119905 gt 0

1 (119909 0) = 11990610 (119909) 2 (119909 0) = 11990620 (119909) V (119909 0) = V0 (119909)

119909 isin Ω

(68)

where max is defined in the same fashion as (32) and thetruncation ℎ(1199061 1199062 V) is defined as follows

ℎ (1199061 1199062 V) fl ℎ (1199061 1199062 V) if V le Vℎ (1199061 1199062 V) if V gt V (69)

Then it is easy to see that100381710038171003817100381711003817100381710038171003817119871infin(Ω) le 1198911 (V)sdotmax ( 12058111 minus 120591 + 1205831) (12058101 + 1205831)minus1 1003817100381710038171003817119906101003817100381710038171003817119871infin(Ω)

100381710038171003817100381721003817100381710038171003817119871infin(Ω) le 1198912 (V)sdotmax ( 12058121 minus 120591 + 1205832) (12058102 + 1205832)minus1 1003817100381710038171003817119906201003817100381710038171003817119871infin(Ω)

(70)

as far as V le VBy analogy with 119879lowast we define

lowast fl max if V le V in (0 max) V if V119871infin(Ωmax )

gt V (71)

Discrete Dynamics in Nature and Society 9

where V lt max satisfies that V119871infin(ΩV ) le V and

V119871infin(ΩV+120575) gt V for any 120575 gt 0 To prove lowast = max wesuppose

lowast lt max (72)

then ℎ(1 2 V) le ℎ(1199061 1199062 V) = ℎ(1199061 1199062 V) lt 0 It follows toeasily check that V is a strict upper-solution to

V119905 minus 120591ΔV = ℎ (1 2 V) in Ω times (0 lowast) 120597V120597] = 0 on 120597Ω times (0 lowast)

V (119909 0) = V0 (119909) in Ω(73)

This implies V(119909 119905) lt V forall119909 isin Ω 119905 isin (0 lowast) whichcontradicts (72) and proves lowast = max as well as max = infin

Thanks to V le V we show that (1 2 V) is also a solutionto (6) So we have

119879lowast = 119879max ge max = infin (74)

The proof of Theorem 3 is completed

Competing Interests

The author declares that they have no competing interests

Acknowledgments

Thiswork is supported by the ChinaNational Natural ScienceFoundation (nos 11361055 and 11261053) and the NaturalScience ResearchKey Project of University of Anhui Province(no KJ2015A251)

References

[1] J D Murray Mathematical Biology vol 19 Springer BerlinGermany 2nd edition 1993

[2] M Winkler ldquoBoundedness in the higher-dimensionalparabolic-parabolic chemotaxis system with logistic sourcerdquoCommunications in Partial Differential Equations vol 35 no 8pp 1516ndash1537 2010

[3] T Hillen and K J Painter ldquoA userrsquos guide to PDE models forchemotaxisrdquo Journal of Mathematical Biology vol 58 no 1-2pp 183ndash217 2009

[4] E F Keller and L A Segel ldquoInitiation of slimemold aggregationviewed as an instabilityrdquo Journal of Theoretical Biology vol 26no 3 pp 399ndash415 1970

[5] M Winkler ldquoAbsence of collapse in a parabolic chemo-taxis system with signal-dependent sensitivityrdquo MathematischeNachrichten vol 283 no 11 pp 1664ndash1673 2010

[6] J I Tello and M Winkler ldquoStabilization in a two-specieschemotaxis system with a logistic sourcerdquo Nonlinearity vol 25no 5 pp 1413ndash1425 2012

[7] M Negreanu and J I Tello ldquoAsymptotic stability of a twospecies chemotaxis systemwith non-diffusive chemoattractantrdquoJournal of Differential Equations vol 258 no 5 pp 1592ndash16172015

[8] Q Zhang and Y Li ldquoGlobal boundedness of solutions toa two-species chemotaxis systemrdquo Zeitschrift fur angewandteMathematik und Physik vol 66 no 1 pp 83ndash93 2015

[9] C Mu L Wang P Zheng and Q Zhang ldquoGlobal existenceand boundedness of classical solutions to a parabolic-parabolicchemotaxis systemrdquo Nonlinear Analysis Real World Applica-tions vol 14 no 3 pp 1634ndash1642 2013

[10] Y Tao ldquoBoundedness in a chemotaxis model with oxygenconsumption by bacteriardquo Journal ofMathematical Analysis andApplications vol 381 no 2 pp 521ndash529 2011

[11] D Horstmann and M Winkler ldquoBoundedness vs blow-up in achemotaxis systemrdquo Journal of Differential Equations vol 215no 1 pp 52ndash107 2005

[12] M Winkler ldquoAggregation vs global diffusive behavior in thehigher-dimensional Keller-Segel modelrdquo Journal of DifferentialEquations vol 248 no 12 pp 2889ndash2905 2010

[13] PQuittner andP Souplet Superlinear Parabolic Problems Blow-Up Global Existence and Steady States Birkhauser AdvancedTexts Birkhauser Basel Switzerland 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

where V lt max satisfies that V119871infin(ΩV ) le V and

V119871infin(ΩV+120575) gt V for any 120575 gt 0 To prove lowast = max wesuppose

lowast lt max (72)

then ℎ(1 2 V) le ℎ(1199061 1199062 V) = ℎ(1199061 1199062 V) lt 0 It follows toeasily check that V is a strict upper-solution to

V119905 minus 120591ΔV = ℎ (1 2 V) in Ω times (0 lowast) 120597V120597] = 0 on 120597Ω times (0 lowast)

V (119909 0) = V0 (119909) in Ω(73)

This implies V(119909 119905) lt V forall119909 isin Ω 119905 isin (0 lowast) whichcontradicts (72) and proves lowast = max as well as max = infin

Thanks to V le V we show that (1 2 V) is also a solutionto (6) So we have

119879lowast = 119879max ge max = infin (74)

The proof of Theorem 3 is completed

Competing Interests

The author declares that they have no competing interests

Acknowledgments

Thiswork is supported by the ChinaNational Natural ScienceFoundation (nos 11361055 and 11261053) and the NaturalScience ResearchKey Project of University of Anhui Province(no KJ2015A251)

References

[1] J D Murray Mathematical Biology vol 19 Springer BerlinGermany 2nd edition 1993

[2] M Winkler ldquoBoundedness in the higher-dimensionalparabolic-parabolic chemotaxis system with logistic sourcerdquoCommunications in Partial Differential Equations vol 35 no 8pp 1516ndash1537 2010

[3] T Hillen and K J Painter ldquoA userrsquos guide to PDE models forchemotaxisrdquo Journal of Mathematical Biology vol 58 no 1-2pp 183ndash217 2009

[4] E F Keller and L A Segel ldquoInitiation of slimemold aggregationviewed as an instabilityrdquo Journal of Theoretical Biology vol 26no 3 pp 399ndash415 1970

[5] M Winkler ldquoAbsence of collapse in a parabolic chemo-taxis system with signal-dependent sensitivityrdquo MathematischeNachrichten vol 283 no 11 pp 1664ndash1673 2010

[6] J I Tello and M Winkler ldquoStabilization in a two-specieschemotaxis system with a logistic sourcerdquo Nonlinearity vol 25no 5 pp 1413ndash1425 2012

[7] M Negreanu and J I Tello ldquoAsymptotic stability of a twospecies chemotaxis systemwith non-diffusive chemoattractantrdquoJournal of Differential Equations vol 258 no 5 pp 1592ndash16172015

[8] Q Zhang and Y Li ldquoGlobal boundedness of solutions toa two-species chemotaxis systemrdquo Zeitschrift fur angewandteMathematik und Physik vol 66 no 1 pp 83ndash93 2015

[9] C Mu L Wang P Zheng and Q Zhang ldquoGlobal existenceand boundedness of classical solutions to a parabolic-parabolicchemotaxis systemrdquo Nonlinear Analysis Real World Applica-tions vol 14 no 3 pp 1634ndash1642 2013

[10] Y Tao ldquoBoundedness in a chemotaxis model with oxygenconsumption by bacteriardquo Journal ofMathematical Analysis andApplications vol 381 no 2 pp 521ndash529 2011

[11] D Horstmann and M Winkler ldquoBoundedness vs blow-up in achemotaxis systemrdquo Journal of Differential Equations vol 215no 1 pp 52ndash107 2005

[12] M Winkler ldquoAggregation vs global diffusive behavior in thehigher-dimensional Keller-Segel modelrdquo Journal of DifferentialEquations vol 248 no 12 pp 2889ndash2905 2010

[13] PQuittner andP Souplet Superlinear Parabolic Problems Blow-Up Global Existence and Steady States Birkhauser AdvancedTexts Birkhauser Basel Switzerland 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of