ARTICLE IN PRESS

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Statistics & Probability Letters 78 (2008) 490–498

www.elsevier.com/locate/stapro

Exponential stability of non-autonomous stochastic partialdifferential equations with finite memory$

Li Wana,b,�, Jinqiao Duana,c

aDepartment of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, ChinabDepartment of Mathematics and Physics, Wuhan University of Science and Engineering, Wuhan 430073, China

cDepartment of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

Received 19 July 2007; received in revised form 13 August 2007; accepted 24 August 2007

Available online 31 August 2007

Abstract

The exponential stability, in both mean square and almost sure senses, for energy solutions to a nonlinear and non-

autonomous stochastic partial differential equations with finite memory is investigated. Various criteria for stability are

obtained. An example is presented to demonstrate the main results.

r 2007 Elsevier B.V. All rights reserved.

MSC: primary 37L55; 35R60; secondary 60H15; 37H20; 34D35

Keywords: Stochastic partial differential equations; Energy solutions; Energy equation; Exponential stability

1. Introduction

Recently stochastic partial differential equations have attracted a lot of attention, and various results on theexistence, uniqueness and the asymptotic behaviors of the solutions have been established; see, for example,Caraballo et al. (2000), Ichikawa (1982), Ichikawa (1983), Kwiecinska (2001), Leha et al. (1999), Liu andMandrekar (1997), Liu (2006), Maslowski (1995) and Waymire and Duan (2005). In particular, stability ofsolutions has been studied by the methods of coercivity conditions, the Lyapunov functionals, and energyestimates; see Caraballo and Liu (1999), Liu and Mao (1998) and Taniguchi (1995).

A few authors have studied stochastic partial differential equations in which the forcing term contains somehereditary features; see, for example, Caraballo et al. (2002a), Caraballo et al. (2000) and Taniguchi et al. (2002).These situations may appear, for instance, when controlling a system by applying a force which takes intoaccount not only the present state of the system but also the history of the solutions. The exponential stability ofthe mild solutions to the semilinear stochastic delay evolution equations was discussed by using Lyapunovfunctionals; see Liu (1998) and Taniguchi (1998). When discussing the asymptotic behavior of solutions, the

e front matter r 2007 Elsevier B.V. All rights reserved.

l.2007.08.003

was supported in part by the Natural Science Foundation of China (No. 10171059) and by the NSF Grant 0620539.

ing author. Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, China.

esses: wanlinju@yahoo.com.cn (L. Wan), duan@iit.edu (J. Duan).

ARTICLE IN PRESSL. Wan, J. Duan / Statistics & Probability Letters 78 (2008) 490–498 491

method by Lyapunov functionals is powerful. However, it is well known that the construction of Lyapunovfunctionals is more difficult for functional differential equations such as differential equations with memory.

The purpose of this paper is to discuss the mean square exponential stability and almost sure exponentialstability of the energy solutions to the following nonlinear and non-autonomous stochastic partial differentialequation with finite memory:

dX ðtÞ ¼ ½Aðt;X ðtÞÞ þ F ðt;X tÞ�dtþ Gðt;X tÞdW ðtÞ; tX0,

X ðsÞ ¼ jðsÞ 2 L2ðO;Cð½�r; 0�;HÞÞ; s 2 ½�r; 0�, ð1Þ

in which C:¼Cð½�r; 0�;HÞ denotes the space of all continuous functions from ½�r; 0� into H;j is F0-measurableand A : ½0;1Þ � V ! V� and F : ½0;1Þ � C! V� and G : ½0;1Þ � C ! L0

QðK ;HÞ are continuous.The contents of this paper are as follows. In Section 2 we present some preliminaries and consider the

existence of energy solutions (see Definition 2.1). In Section 3 we consider stability of the nonlinear and non-autonomous stochastic partial differential equations with finite memory. In Section 4 we present an examplewhich illustrates the main results in this paper.

2. Preliminaries

Let V, H and K be separable Hilbert spaces and let LðK ;HÞ be the space of all bounded linear operator fromK to H. We denote the norms of elements in V ;H;K and LðV ;HÞ by k � k; j � j2; j � jK and j � j, respectively. Andj � j� denotes the norm of V�; h�; �i denotes the duality between V and V�. Suppose that V and H satisfy

V � H � H� � V�,

where V is a dense subspace of H and the injections are continuous with

l1jvj22pkvk2; l140; v 2 V . (2)

We are given a Q-Wiener process in the complete probability space ðO;F;P; fFtgtX0Þ and have values in K,i.e. W ðtÞ is defined as

W ðtÞ ¼X1n¼1

ffiffiffiffiffiln

pBnðtÞen; tX0,

where BnðtÞ; ðn ¼ 1; 2; . . .Þ is a sequence of real value standard Brownian motions mutually independent onðO;F;P; fFtgtX0Þ; lnX0; ðn ¼ 1; 2; . . .Þ are non-negative real numbers such that

PnX1lno1; fengnX1 is a

complete orthonormal basis in K, and Q 2LðK ;KÞ is the incremental covariance operator of the processW ðtÞ, which is a symmetric non-negative trace class operator defined as

Qen ¼ lnen; n ¼ 1; 2; . . . .

Let L0QðK ;HÞ be the space of all bounded linear operators from K to H with the following condition:

kxk2L0

Q

¼ trðxQx�Þo1; x 2 LðK ;HÞ.

Let M2ð�r;T ;V Þ denote the space of all V-valued measurable functions defined on ½�r;T � � O withER T

�rkX ðtÞk2 dto1.

First, we give the definition of an energy solution to (1).

Definition 2.1. Stochastic process X ðtÞ on ðO;F;P; fFtgtX0Þ is called an energy solution to (1) if the followingconditions are satisfied:

(i)

X ðtÞ 2M2ð�r;T ;V Þ \ L2ðO;Cð�r;T ;HÞÞ;T40, (ii) the following equation holds in V� almost surely, for t 2 ½0;TÞ,

X ðtÞ ¼ X ð0Þ þ

Z t

0

½Aðs;X ðsÞÞ þ F ðs;X sÞ�dsþ

Z t

0

Gðs;X sÞdW ðsÞ; tX0,

X ðsÞ ¼ jðsÞ; s 2 ½�r; 0�,

ARTICLE IN PRESSL. Wan, J. Duan / Statistics & Probability Letters 78 (2008) 490–498492

the following stochastic energy equality holds:

(iii)

jX ðtÞj22 ¼ jX ð0Þj22 þ 2

Z t

0

hX ðsÞ;Aðs;X ðsÞÞ þ F ðs;X sÞidsþ

Z t

0

kGðs;X sÞk2L0

Q

ds

þ 2

Z t

0

hX ðsÞ;Gðs;X sÞdW ðsÞi. ð3Þ

In order to guarantee the existence and uniqueness of energy solution X ðtÞ to (4), we need the followingconditions:

(A1)

(Monotonicity and coercivity) There exist a40 and l 2 R such that for a.e. t 2 ð0;TÞ,

�2hAðt; uÞ � Aðt; vÞ; u� vi þ lju� vj22Xaku� vk2.

(A2)

(Measurability) For any v 2 V , the mapping t 2 ð0;TÞ ! Aðt; vÞ 2 V� is measurable. (A3) (Hemicontinuity) The next mapping is continuous for any u; v;w 2 V , a.e. t 2 ð0;TÞ:

m 2 R! hAðt; uþ mvÞ;wi 2 R.

(A4)

(Boundedness) There exists c40 such that for any v 2 V , a.e. t 2 ð0;TÞ:

jAðt; vÞj�pckvk.

(A5)

(Lipschitz condition) There exists c140 such that for any x; Z 2 C and F ðt; 0Þ 2 L2ð½�r; 0� � O;V�Þ andGðt; 0Þ 2 L2ð½�r; 0� � O;HÞ,

kF ðt; xÞ � F ðt; ZÞk�pc1jx� ZjC ; kGðt; xÞ � Gðt; ZÞkL0Qpc1jx� ZjC .

We have the following result on the energy solution to (4); see, for example, Caraballo et al. (2002a,b),Liu (2006) and Pardoux (1975).

Theorem 2.2. Suppose that conditions (A1)–(A5) are satisfied. Then there exists a unique energy solution X ðtÞ to

(4). Furthermore, the following identity holds:

d

dtEjX ðtÞj22 ¼ 2EhX ðsÞ;Aðt;X ðtÞÞ þ F ðt;X tÞi þ EkGðt;X tÞk

2L0

Q

; tXt0.

Throughout this paper, we assume the existence of the energy solutions to (1) with Ekjk2Co1.

3. Main results

In this section we discuss the stochastic partial differential equations with finite memory which are theexamples of (1). Let r; t : ½0;1Þ ! ½0; r� be continuous functions and r40 is a constant. Assume that A :½0;1Þ � V ! V� and f : ½0;1Þ ! V� are Lebesgue measurable. Let g : ½0;1Þ �H ! H and h : ½0;1Þ �H ! LQðK ;HÞ be Lipschitz continuous uniformly in t.

We consider the following stochastic partial differential equation with finite memory:

dX ðtÞ ¼ ½Aðt;X ðtÞÞ þ f ðtÞ�dtþ gðt;X ðt� rðtÞÞdtþ hðt;X ðt� tðtÞÞdW ðtÞ; tX0, (4)

with the initial condition

X ðsÞ ¼ jðsÞ 2 L2ðO;Cð½�r; 0�;HÞÞ; s 2 ½�r; 0�,

ARTICLE IN PRESSL. Wan, J. Duan / Statistics & Probability Letters 78 (2008) 490–498 493

where f 2 L2ð½0;1Þ;V�Þ. Set F 1ðt;cÞ ¼ gðt;cð�rðtÞÞÞ and Gðt;cÞ ¼ hðt;cð�tðtÞÞÞ for any c 2 C. Then (4) canbe viewed as a stochastic functional partial differential equation (1) with F ðt;cÞ ¼ F1ðt;cÞ þ f ðtÞ. Note thatwe do not require that rðtÞ and tðtÞ are differentiable functions.

Now we first state the following result on exponential stability in mean square.

Theorem 3.1. Suppose that Eq. (4) satisfies the following conditions:

(B1)

conditions (A2)–(A4) hold and there exist d140 and a continuous, integrable function a1ðtÞ40 such that for

a.e. t 2 ½0;1Þ; u; v 2 V ,

�2hAðt; uÞ � Aðt; vÞ; u� vi þ a1ðtÞju� vj22Xd1ku� vk2;

(B2)

there exist integrable functions a2;b2 : ½0;1Þ ! Rþ such that

jgðt; uÞj22pðd2 þ a2ðtÞÞjuj22 þ b2ðtÞ for d2X0; u 2 H;

(B3)

there exist integrable functions a3;b3 : ½0;1Þ ! Rþ such that

khðt; uÞk2L0

Q

pðd3 þ a3ðtÞÞjuj22 þ b3ðtÞ for d3X0; u 2 H;

(B4)

there exists s140 such that

Z 10

es1tjf bðtÞj2� dto1;

Z 10

es1tbiðtÞdto1; i ¼ 2; 3;

(B5)

d1l142ffiffiffiffiffid2pþ d3, where l1 is defined by (2).

Then for any energy solution X ðtÞ to (4), there exist s 2 ð0;s1Þ and B41 such that

EjX ðtÞj22pBe�st; tX0. (5)

In other words the energy solution X ðtÞ to (4) converges to zero exponentially in mean square as t!1.

Proof. From (B5), there exists g140 such that

ðd1 � g1Þl142ffiffiffiffiffid2

pþ d3.

Thus we can take g240 such that

ðd1 � g1Þl14g2 þd2g2þ d342

ffiffiffiffiffid2

pþ d3.

Furthermore, we can choose s 2 ð0;s1Þ such that

ðd1 � g1Þl14sþ g2 þ esr d2g2þ esrd3.

Now define the function f 1 : ½0;1Þ � V ! V� by

f 1ðt; vÞ ¼ Aðt; vÞ þ f ðtÞ; v 2 V ; tX0.

ARTICLE IN PRESSL. Wan, J. Duan / Statistics & Probability Letters 78 (2008) 490–498494

From (B1) and f 2 L2ð½0;1Þ;V�Þ, it follows that

2hf 1ðt; vÞ; vipa1ðtÞjvj22 � d1kvk2 þ 2hf ðtÞ; vi

pa1ðtÞjvj22 � d1kvk2 þ g1kvk2 þ g�11 jf ðtÞj

2�

p½ð�d1 þ g1Þl1 þ a1ðtÞ�jvj22 þ g�11 jf ðtÞj2�

¼ ½�aþ a1ðtÞ�jvj22 þ b1ðtÞ; v 2 V ,

where a ¼ ðd1 � g1Þl1;b1ðtÞ ¼ g�11 jf ðtÞj2�.

Set

yðtÞ ¼ a1ðtÞ þ elr a2ðtÞg2þ elra3ðtÞ; bðtÞ ¼ b1ðtÞ þ

b2ðtÞg2þ b3ðtÞ.

It follows from (B1)–(B4) and f 2 L2ð½0;1Þ;V�Þ that

R1 ¼

Z 10

yðsÞdso1; R2 ¼

Z 10

bðsÞdspR3 ¼

Z 10

es1sbðsÞdso1.

Set

KðtÞ ¼EjX ðtÞj22e

st exp �R t

0½yðsÞ þ essbðsÞ�ds

� �; tX0;

EjX ðtÞj22est; �rpto0:

((6)

It is clear that KðtÞ is continuous on ½�r;1Þ and

dKðtÞ

dt¼ est exp �

Z t

0

½yðsÞ þ essbðsÞ�ds

� �fsEjX ðtÞj22 � ½yðtÞ þ estbðtÞ�EjX ðtÞj22

þ 2Ehf 1ðt;X ðtÞÞ;X ðtÞi þ 2Ehgðt;X ðt� rðtÞÞÞ;X ðtÞi þ Ekhðt;X ðt� tðtÞÞÞk2L0

Q

g. ð7Þ

From (B2) and (B3), it follows that for tX0

dKðtÞ

dtpest exp �

Z t

0

½yðsÞ þ essbðsÞ�ds

� �sEjX ðtÞj22 � ½yðtÞ þ estbðtÞ�EjX ðtÞj22

�þ ½�aþ a1ðtÞ�EjX ðtÞj22

þ b1ðtÞ þ g2EjX ðtÞj22 þEjgðt;X ðt� rðtÞÞÞj22

g2þ Ekhðt;X ðt� tðtÞÞÞk2

L0Q

�p½�aþ a1ðtÞ þ sþ g2 � yðtÞ�KðtÞ þ estbðtÞ � estbðtÞKðtÞ

þ est exp �

Z t

0

½yðsÞ þ essbðsÞ�ds

� �d2 þ a2ðtÞ

g2EjX ðt� rðtÞÞj22

þ est exp �

Z t

0

½yðsÞ þ essbðsÞ�ds

� �ðd3 þ a3ðtÞÞEjX ðt� tðtÞÞj22. ð8Þ

Now we claim that

KðtÞp1þ supt2½�r;0�

fEjX ðtÞj22g ¼M for all tX0. (9)

In fact, if (9) is false, then there exists t140 such that for 8�40,

KðtÞpM ; 0ptot1; Kðt1Þ ¼M ; KðtÞ4M; t1otot1 þ �. (10)

Since dKðtÞ=dt exists (by (7)), we see that

d

dtKðt1ÞX0. (11)

ARTICLE IN PRESSL. Wan, J. Duan / Statistics & Probability Letters 78 (2008) 490–498 495

From (8), it follows that

d

dtKðt1Þp½�aþ a1ðt1Þ þ sþ g2 � yðt1Þ�Kðt1Þ

þ est1 exp �

Z t1

0

½yðsÞ þ essbðsÞ�ds

� �d2 þ a2ðt1Þ

g2EjX ðt1 � rðt1ÞÞj22

þ est1 exp �

Z t1

0

½yðsÞ þ essbðsÞ�ds

� �ðd3 þ a3ðt1ÞÞEjX ðt1 � tðt1ÞÞj22. ð12Þ

We consider the following three different cases:

(i)

Suppose that t1 � rðt1ÞX0; t1 � tðt1ÞX0. From (10) and (12), it follows that

d

dtKðt1Þp½�aþ a1ðt1Þ þ sþ g2 � yðt1Þ�Kðt1Þ

þ esrðt1Þ exp �

Z t1

t1�rðt1Þ½yðsÞ þ essbðsÞ�ds

� �d2 þ a2ðt1Þ

g2Kðt1 � rðt1ÞÞ

þ estðt1Þ exp �

Z t1

t1�tðt1Þ½yðsÞ þ essbðsÞ�ds

� �ðd3 þ a3ðt1ÞÞKðt1 � tðt1ÞÞ

p½�aþ a1ðt1Þ þ sþ g2 � yðt1Þ�M þ esr d2 þ a2ðt1Þg2

M þ esrðd3 þ a3ðt1ÞÞM

p �aþ sþ g2 þ esr d2g2þ esrd3

Mo0.

(ii)

Suppose that t1 � rðt1Þo0; t1 � tðt1ÞX0. Then t1 � rðt1Þ4� r. From (10) and (12), it follows that

d

dtKðt1Þp½�aþ a1ðt1Þ þ sþ g2 � yðt1Þ�Kðt1Þ

þ esrðt1Þ exp �

Z t1

0

½yðsÞ þ essbðsÞ�ds

� �d2 þ a2ðt1Þ

g2Kðt1 � rðt1ÞÞ

þ estðt1Þ exp �

Z t1

t1�tðt1Þ½yðsÞ þ essbðsÞ�ds

� �ðd3 þ a3ðt1ÞÞKðt1 � tðt1ÞÞ

p �aþ sþ g2 þ esr d2g2þ esrd3

Mo0.

(iii)

Suppose that t1 � rðt1Þo0; t1 � tðt1Þo0. Then t1 � rðt1Þ4� r; t1 � tðt1Þ4� r. Similarly, it followsthat

d

dtKðt1Þp �aþ sþ g2 þ esr d2

g2þ esrd3

Mo0.

Thus, one obtains

d

dtKðt1Þo0,

which contradicts with (11). Hence, (9) holds true and we obtain

EjX ðtÞj22pe�st exp

Z t

0

½yðsÞ þ essbðsÞ�ds

� �Mpe�stB; tX0,

where B ¼ eR1þR3M. The proof is complete. &

ARTICLE IN PRESSL. Wan, J. Duan / Statistics & Probability Letters 78 (2008) 490–498496

Remark 3.2. Comparing with Taniguchi (2007), our method does not assume that r and t are differentiablefunctions. Hence, our method can be applied to more general stochastic partial differential equations withmemory.

Next, we state the result of almost sure exponential stability.

Theorem 3.3. Suppose that all the conditions of Theorem 3.1 are satisfied. If the following additional condition is

satisfied:

(B6)

Both aiðtÞ and estbiðtÞ ði ¼ 1; 2; 3Þ are bounded functions, where b1ðtÞ ¼ g�11 jf ðtÞj2�.

Then there exists TðoÞ40 such that for all t4TðoÞ,

jX ðtÞj22pes=2e�st=2,

with probability one.

Proof. Let N1 and N2 be positive integers such that

N1 � rðN1ÞXN1 � rX1; N2 � tðN2ÞXN2 � rX1.

Let N4N3 ¼ maxðN1;N2Þ and IN ¼ ½N;N þ 1�.Set

aðtÞ ¼ a1ðtÞ þ g2 þd2 þ a2ðtÞ

g2esr þ 32ðd3 þ a3ðtÞÞesr,

bðtÞ ¼ 2b1ðtÞ þ2b2ðtÞg2þ 64b3ðtÞ.

It follows from (B6) that there exists B140 such that

aðtÞ þ estbðtÞpB1.

Then we obtain from (3)

E supt2IN

jX ðtÞj22pEjX ðNÞj22 þ 2E supt2IN

Z t

N

hX ðsÞ;Aðs;X ðsÞÞ þ f ðsÞidsþ 2E supt2IN

Z t

N

hX ðsÞ; gðt;X ðs� rðsÞÞÞids

þ E supt2IN

Z t

N

khðt;X ðs� tðsÞÞÞk2L0

Q

dsþ 2E supt2IN

Z t

N

hX ðsÞ; hðt;X ðs� tðsÞÞÞdW ðsÞi.

Then, by the Burkholder–Davis–Gundy inequality (Mao, 1994; Revuz and Yor, 1999),

2E supt2IN

Z t

N

hX ðsÞ; hðt;X ðs� tðsÞÞÞdW ðsÞip1

2E sup

t2IN

jX ðtÞj22 þ 32

Z Nþ1

N

Ekhðt;X ðs� tðsÞÞÞkL0Qds.

Therefore it follows from (B1)–(B3) and (5) that

E supt2IN

jX ðtÞj22p2EjX ðNÞj22 þ 4E supt2IN

Z t

N

hX ðsÞ;Aðs;X ðsÞÞ þ f ðsÞids

þ 4E supt2IN

Z t

N

hX ðsÞ; gðt;X ðs� rðsÞÞÞids

ARTICLE IN PRESSL. Wan, J. Duan / Statistics & Probability Letters 78 (2008) 490–498 497

þ 64

Z Nþ1

N

Ekhðt;X ðs� tðsÞÞk2L0

Q

ds

p2EjX ðNÞj22 þ 2

Z Nþ1

N

a1ðsÞEjX ðsÞj22 þ b1ðsÞds

þ 2

Z Nþ1

N

g2EjX ðsÞj22 þ

d2 þ a2ðsÞg2

EjX ðs� rðsÞÞj22 þb2ðsÞg2

ds

þ 64

Z Nþ1

N

ðd3 þ a3ðsÞÞEjX ðs� tðsÞÞj22 þ b3ðsÞds

p2Be�sN þ

Z Nþ1

N

2Be�ssfaðsÞ þ bðsÞessgdsp2Be�sN 1þB1

s

� �.

Let �N be any fixed positive real number. Then,

P supt2IN

jX ðtÞj224�2N

� �p

Esupt2INjX ðtÞj22

�2Np

2Be�sN 1þB1

s

� ��2N

.

For each integer N , choosing �2N ¼ e�sN=2, then

P supt2IN

jX ðtÞj224e�sN=2

� �p2Be�sN=2 1þ

B1

s

� �.

From the Borel–Cantelli lemma (Ash, 2000; Mao, 1994), it follows that there exists TðoÞ40 such thatfor all t4TðoÞ

jX ðtÞj22pes=2e�st=2 a.s.

The proof is complete. &

4. An example

In this section we present an example which illustrates the main results. We consider a sufficient conditionfor any energy solution to a stochastic heat equation with finite memory to converge to zero exponentially inmean square and almost surely exponentially. Let A ¼ g1ðq

2=qx2Þ, where g140;H ¼ L2ð0; pÞ and

H10 ¼ u 2 L2ð0;pÞ:

qu

qx2 L2ð0;pÞ; uð0Þ ¼ uðpÞ ¼ 0

� �,

H2 ¼ u 2 L2ð0;pÞ :qu

qx;q2

qx22 L2ð0;pÞ; uð0Þ ¼ uðpÞ ¼ 0

� �.

Operator A has the domain DðAÞ ¼ H10 \H2. Define the norms of two spaces H and H1

0 by jxj22 ¼

R p0x2ðxÞdx

for any x 2 H and kuk2 ¼R p0 ðqu=qxÞ2 dx for any u 2 H1

0, respectively. Then it is known that

hAu; uip� g1kuk2; u 2 H1

0.

Let rðtÞ and tðtÞ be the non-differentiable function defined by

rðtÞ ¼1

1þ j sin tj; tðtÞ ¼

1

1þ j cos tj; tX0.

Consider the following stochastic heat equation with finite memory rðtÞ and tðtÞ:

dX ðtÞ ¼ AX ðtÞ þ gðt;X ðt� rðtÞÞÞdtþ hðt;X ðt� tðtÞÞÞ dwðtÞ, (13)

X ðt; 0Þ ¼ X ðt;pÞ ¼ 0; tX0; X ðs; xÞ ¼ jðs;xÞ; s 2 ½�1=2; 0�;x 2 ½0;p�,

j 2 Cð½�1=2; 0�;L2ð0;pÞÞ; kjkCo1,

ARTICLE IN PRESSL. Wan, J. Duan / Statistics & Probability Letters 78 (2008) 490–498498

where wðtÞ is the one-dimensional standard Wiener process,

gðt; yÞ ¼ ðb1 þ k1ðtÞÞyþ e�ktp; hðt; yÞ ¼ ðb2 þ k2ðtÞÞy,

for any y 2 H; tX0, p 2 H with jpj2o1; k1; k2 : ½0;1Þ ! Rþ are continuous functions, b1; b2 and k arepositive real numbers. It is easy to obtain

jgðt; yÞj22p4ðb21 þ k2

1ðtÞÞjyj22 þ 2e�2ktjpj22,

khðt; yÞk2L0

Q

p4ðb22 þ k2

2ðtÞÞjyj22.

Note that l1 ¼ 1; d1 ¼ 2g1; d2 ¼ 4b21; d3 ¼ 4b2

2; a2ðtÞ ¼ 4k21ðtÞ; a3ðtÞ ¼ 4k2

2ðtÞ;b2ðtÞ ¼ 2e�2ktjpj22;b3ðtÞ ¼ 0 andtaking a1ðtÞ ¼ 1;s1 ¼ 2k. Now we suppose that

2g144b1 þ 4b22

and k21ðtÞ; k

22ðtÞ are decreasing, bounded and integrable functions. Then, it follows from Theorems 3.1

and 3.3 that any energy solution X ðtÞ to (13) converges to zero exponentially in mean square and almost surelyexponentially as t!1.

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