Commun Nonlinear Sci Numer Simulat 19 (2014) 1433–1441

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Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier .com/locate /cnsns

Random attractor for stochastic lattice dynamical systemswith a-stable Lévy noises q

1007-5704/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cnsns.2013.08.036

q This work has been partially supported by NSFC Grants 11071165, 11161015 and 11071199, NSF of Guangxi Grants 2013GXNSFBA019008,Provincial Department of Research Project Grants 2013YB102, the Program to Sponsor Teams for Innovation in the Construction of Talent HighGuangxi Institutions of Higher Learning Grants [2011]47.⇑ Corresponding author. Tel.: +86 0773 5896179.

E-mail address: gahui@glut.edu.cn (A. Gu).

Anhui Gu ⇑, Wu AiCollege of Science, Guilin University of Technology, Guilin, Guangxi 541004, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 April 2012Received in revised form 26 August 2013Accepted 27 August 2013Available online 5 September 2013

Keywords:Stochastic lattice dynamical systemsRandom attractora-Stable Lévy noises

The present paper is devoted to the existence of a random attractor for stochastic latticedynamical systems with a-stable Lévy noises.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction and motivation

In this paper, we consider the long-term behavior of the following stochastic lattice Marcus canonical equation:

duiðtÞdt

¼ mðui�1 � 2ui þ uiþ1Þ � kui � fiðuiÞ þ gi þXN

j¼1

rjui}dLj

t

dt; i 2 Z; ð1:1Þ

where u ¼ ðuiÞi2Z 2 ‘2;Z denotes the integers set, m and k are positive constants, fi is a smooth function satisfying some dis-

sipative conditions, g ¼ ðgiÞi2Z 2 ‘2;rj 2 R for j ¼ 1; . . . ;N; Lj

t are mutually independent a-stable Lévy motions (1 < a < 2),and } denotes the Marcus sense in the stochastic term.

Recently, the dynamics of deterministic lattice dynamical systems has drawn much attention of mathematicians andphysicists, see e.g., [1–5] and the references therein. Since most of the realistic systems involve noises which may play animportant role as intrinsic phenomena rather than just compensation for defects in deterministic models, stochastic latticedynamical systems (SLDS) then arise naturally wherein these random influences or uncertainties are taken into account.Since Bates et al. [6] initiated the study of SLDS, there have been many publications regarding the existence of global randomattractors for SLDS with white noises on lattices Z (see e.g., [7–11]). Later, the existence of global random attractors was ex-tended to other SLDS with additive white noises, for example, first-order SLDS on Zk [8], stochastic Ginzburg–Landau latticeequations [9], stochastic FitzHugh–Nagumo lattice equations [10], second-order stochastic lattice systems [11] and first (orsecond)-order SLDS with a multiplicative white noise [7,12]. Zhao and Zhou [13] gave some sufficient conditions for the

Guangxilands in

1434 A. Gu, W. Ai / Commun Nonlinear Sci Numer Simulat 19 (2014) 1433–1441

existence of a global random attractor for general SLDS in the non-weighted space R of infinite sequences and provided anapplication to damped Sine–Gordon lattice systems with additive noises. Very recently, Han, et al. [14] provided some suf-ficient conditions for the existence of global compact random attractors for general random dynamical systems in theweighted space ‘p

q ðp P 1Þ of infinite sequences, and their results are applied to second-order SLDS in [15]. The existenceof a global compact random attractor for SLDS driven by fractional Brownian motions was discussed in [16].

However, all the works above are considered in the framework of Gaussian noises (in terms of Brownian and fractionalBrownian motions). There are no results on these systems when they are perturbed by a non-Gaussian noise (in terms ofLévy noise) to the best of our knowledge. Gaussian processes like Brownian motion have been widely used to model fluctu-ations in engineering and science. The sample paths of a particle driven by Brownian motion are continuous in time almostsurely (i.e., no jumps), the mean square displacement increases linearly in time (i.e., normal diffusion), and the probabilitydensity function decays exponentially in space (i.e., light tail or exponential relaxation) [17]. But some complex phenomenainvolve non-Gaussian fluctuations with peculiar properties such as anomalous diffusion (mean square displacement is anonlinear power law of time) [18] and heavy tail distribution (non-exponential relaxation) [19]. For this topic, we can referto [20–23] for more details. A Lévy motion Lt is a non-Gaussian process with independent and stationary increments, i.e.,increments DLt ¼ LtþDt � Lt are stationary and independent for any non overlapping time lags Dt. Moreover, its sample pathsare only continuous in probability, namely, PðjLt � Lt0 jP dÞ ! 0 as t ! t0 for any positive d. With a suitable modification,these path may be taken as càdlàg, i.e., paths are continuous on the right and have limits on the left. This continuity is weakerthan the usual continuity in time. Indeed, a càdlàg function has at most countably many discontinuities on any time interval,which generalizes the Brownian motion to some extent (see e.g., [24]). As a special case of Lévy processes, the symmetric a-stable Lévy motion plays an important role among stable processes just like Brownian motion among Gaussian processes. Forthe definition of a-stable Lévy motions (0 < a < 2), see [25]. It is worth mentioning that when a ¼ 2, we have the standardBrownian motion, which the Marcus integral (see e.g., [26]) reduces to the Stratonovich stochastic integral and the existenceof a random attractor for Eq. (1.1) has been considered in [7]. For the further development on Lévy motions, we can refer tothe recent books [24,27].

The goal of this article is to establish the existence of a random attractor for SLDS with the nonlinearity fi under somedissipative conditions and driven by a-stable Lévy noises with a 2 ð1;2Þ. Following [28], we first introduce an Ornstein–Uhlenbeck process with a stationary solution to transform Eq. (1.1) into a conjugated random integral equation (with a solu-tion in the sense of Carathéodory [29]). We assume here that 1 < a < 2 since this is the only case where the solutions of theOrnstein–Uhlenbeck equations for the a-stable Lévy noises are stationary, which is crucial to our purpose. The case 0 < a < 1presents new challenges for future research.

The paper is organized as follows. In Section 2, we recall some basic concepts in random dynamical systems. In Section 3,we give a unique solution to Eq. (1.1) and make sure that the solution generates a random dynamical system. We establishthe main result, that is, the existence of a random attractor generated by Eq. (1.1) in Section 4.

2. Random dynamical systems

For the reader’s convenience, we introduce some basic concepts related to random dynamical systems and random attrac-tors, which are taken from [14,30,31]. Let ðE; k � kEÞ be a separable Hilbert space and ðX;F ;PÞ be a probability space.

Definition 2.1. A stochastic process fuðt;xÞgtP0;x2X is a continuous random dynamical system (RDS) over ðX;F ;P; ðhtÞt2RÞ ifu is ðB½0;1Þ � F � BðEÞ;BðEÞÞ-measurable, and for all x 2 X,

(i) the mapping uðt;xÞ : E # E, x # uðt;xÞx is continuous for every t P 0.(ii) uð0;xÞ is the identity on E.

(iii) (cocycle property) uðsþ t;xÞ ¼ uðt; hsxÞuðs;xÞ for all s; t P 0.

Definition 2.2.

(i) A set-valued mapping x # BðxÞ � E (we may write it as BðxÞ for short) is said to be a random set if the mapping x #

distEðx;BðxÞÞ is measurable for any x 2 E, where distEðx;DÞ is the distance in E between the element x and the setD � E.

(ii) A random set BðxÞ is said to be bounded if there exist x0 2 E and a random variable rðxÞ > 0 such thatBðxÞ � fx 2 E : kx� x0kE 6 rðxÞ; x0 2 Eg for all x 2 X.

(iii) A random set BðxÞ is called a compact random set if BðxÞ is compact for all x 2 X.(iv) A random bounded set BðxÞ � E is called tempered with respect to ðhtÞt2R if for a.e. x 2 X,

limt!þ1e�ctdðBðh�txÞÞ ¼ 0 for all c > 0, where dðBÞ ¼ supx2BkxkH . A random variable x # rðxÞ 2 R is said to be tem-pered with respect to ðhtÞt2R if for a.e. x 2 X, limt!þ1supt2Re�ctrðh�txÞ ¼ 0 for all c > 0.

We consider an RDS fuðt;xÞgtP0;x2X over ðX;F ;P; ðhtÞt2RÞ and DðEÞ the set of all tempered random sets of E.

A. Gu, W. Ai / Commun Nonlinear Sci Numer Simulat 19 (2014) 1433–1441 1435

Definition 2.3. A random setK is called an absorbing set inDðEÞ if for all B 2 DðEÞand a.e.x 2 X there exists tBðxÞ > 0 such that

uðt; h�txÞBðh�txÞ � KðxÞ for all t P tBðxÞ:

Definition 2.4. A random set A is called a global random DðEÞ attractor (pullback DðEÞ attractor) for fuðt;xÞgtP0;x2X if thefollowing hold:

(i) A is a random compact set, i.e., x # dðx;AðxÞÞ is measurable for every x 2 E and AðxÞÞ is compact for a.e. x 2 X.(ii) A is strictly invariant, i.e., for x 2 X and all t P 0, uðt;xÞAðxÞ ¼ AðhtxÞ.

(iii) A attracts all sets in DðEÞ, i.e. for all B 2 DðEÞ and a.e. x 2 X, we have

limt!þ1

dðuðt; h�txÞBðh�txÞ;AðxÞÞ ¼ 0;

where dðX;YÞ ¼ supx2X infy2Ykx� ykE is the Hausdorff semi-metric (X # E;Y # E).

Definition 2.5. An RDS fuðt;x; �ÞgtP0;x2X defined on a metric dynamical system ðX;F ;P; ðhtÞt2RÞwith state space ‘2 is said tobe asymptotically null in Dð‘2Þ, if for a.e. x 2 X;BðxÞ 2 Dð‘2Þ, and any � > 0, there exist Tð�;x;BðxÞÞ > 0 andMð�;x;BðxÞÞ 2 N such that

X

jij>Mð�;x;BðxÞÞjðuiðt; h�tx;uðh�txÞÞÞj2 6 �; 8t P Tð�;x;BðxÞÞ; 8uðxÞ 2 BðxÞ:

Proposition 2.6. [See [14]] Suppose that

(a) there exists a random bounded absorbing set KðxÞ 2 Dð‘2Þ;x 2 X, such that for any BðxÞ 2 Dð‘2Þ and all x 2 X, thereexists Tðx;BÞ > 0 yielding uðt; h�tx;Bðh�txÞÞ � KðxÞ for all t P Tðx; BÞ;(b) the RDS fuðt;xÞgtP0;x2X is random asymptotically null on KðxÞ, i.e., for any � > 0, there exist Tð�;x;KÞ > 0 andI0ð�;x;KÞ 2 N such that

supu2KðxÞ

Xjij>I0ð�;x;KðxÞÞ

jðuiðt; h�tx; uðh�txÞÞÞj2 6 �2;8t P Tð�;x;KðxÞÞ: ð2:1Þ

Then the RDS fuðt;x; �ÞgtP0;x2X possesses a unique global random DðEÞ attractor given by

eAðxÞ ¼ \sPTðx;KÞ

[tPs

uðt; h�tx;Kðh�txÞÞ: ð2:2Þ

3. SLDS with a-stable Lévy noises

Let ðX;F ;PÞ be a probability space, where X ¼ SðR; ‘2Þ with Skorokhod metric as the canonical sample space of càdlàgfunctions defined on R and taking values in ‘2, F :¼ BðSðR; ‘2ÞÞ the associated Borel r-field and P is the corresponding(Lévy) probability measure on F which is given by the distribution of a two-sided Lévy process with paths in SðR; ‘2Þ, i.e.,xðtÞ ¼ LtðxÞ. Let htxð�Þ ¼ xð� þ tÞ �xðtÞ; t 2 R, then the mapping ðt;xÞ ! htx is continuous and measurable (see [30]),and the (Lévy) probability measure is h-invariant, i.e., Pðh�1

t ðTÞÞ ¼ PðTÞ for all T 2 F (see [24]).We note that Eq. (1.1) is interpreted as a system of integral equations

uiðtÞ ¼ uið0Þ þZ t

0ðmðui�1ðsÞ � 2uiðsÞ þ uiþ1ðsÞÞ � kuiðsÞ � fiðuiðsÞÞ þ giÞdsþ

XN

j¼1

Z t

0rjuiðsÞ}dLj

t ; i 2 Z; ð3:1Þ

where the stochastic integral is understood to be in the Marcus sense.Assumptions on the nonlinearity fi. Let fi 2 CðRÞ satisfy the conditions that supi2Zjf 0ðuÞj is bounded for u in bounded sets and

fiðxÞx P 0 for all x 2 R. Let ~f be the Nemytski operator associated with fi, for u ¼ ðuiÞi2Z 2 ‘2, then ~f ðuÞ 2 ‘2 and ~f is locally

Lipschitz from ‘2 to ‘2 (see [6,7]). In the sequel, when no confusion arises, we identify ~f with f.For convenience, we now formulate system (1.1) as a stochastic differential equation in ‘2. Denote by k � k the norm in the

space ‘2, and for x ¼ ðxiÞi2Z 2 ‘2, define A; eB; eB� to be linear operators from ‘2 to ‘2 as follows:

ðAxÞi ¼ �xi�1 þ 2xi � xiþ1; ðeBxÞi ¼ xiþ1 � xi; ðeB�xÞi ¼ xi�1 � xi; i 2 Z:

It is easy to show that A ¼ eBeB� ¼ eB�eB; heB�x; x0i ¼ hx; eBx0i for all x; x0 2 ‘2, which implies that hAx; xiP 0. Then system (1.1)with initial values u0 ¼ ðu0;iÞi2Z 2 ‘

2, can be written as an equation in ‘2 for t 2 R and x 2 X,

1436 A. Gu, W. Ai / Commun Nonlinear Sci Numer Simulat 19 (2014) 1433–1441

uðtÞ ¼ u0 þZ t

0ð�mAuðsÞ � kuðsÞ � f ðuðsÞÞ þ gÞdsþ

XN

j¼1

Z t

0rjuðsÞ}dLj

t : ð3:2Þ

To prove that this stochastic Eq. (3.2) generates a random dynamical system, we will transform it into a random differentialequation in ‘2.

Now, we introduce the Ornstein–Uhlenbeck processes in ‘2 on the metric dynamical system ðX;F ;P; ðhtÞt2RÞ given by therandom variable

zðhtxÞ ¼ �kZ 0

�1ekshtxðsÞds; t 2 R;x 2 X: ð3:3Þ

The above integrals exist in the sense of any path with a subexponential growth, and z solves the following Ornstein–Uhlen-beck equation

dzþ zdt ¼ dLt ; t 2 R: ð3:4Þ

In fact, we have

Lemma 3.1. Let Lt be a two-sided symmetric a-stable Lévy motion on ‘2 with 1 < a < 2. Then

(i) there exists a fhtgt2R-invariant subset �X 2 F of full measure for a.e. x 2 �X, the random variable

zðxÞ ¼ �kZ 0

�1eksxðsÞds;

is well defined and the unique stationary solutions of (3.4) is given by (3.3). Moreover, the mapping t ! zðhtxÞ is càdlàg;(ii) for x 2 �X, the sample paths xðtÞ of Lt satisfy

limt!�1

xðtÞt¼ 0; t 2 R:

In addition,

limt!�1

jzðhtxÞjjtj ¼ lim

t!�1

1t

Z t

0zðhtxðsÞÞds ¼ 0:

Proof. Due to k > 0 and the sample paths of an a-stable Lévy motion satisfy

lim supt!1

t�1j sup

06s6tkLsk ¼ 0 a:s: or ¼ 1 a:s:

according to whether j < a or j > a (see [25]). Therefore, the proof is analogous to that of Lemma 2 in [28], so we omit thedetails here. h

Let zj be the associated Ornstein–Uhlenbeck process corresponding to (3.4) with Ljt instead of Lt and denote

TðxÞ ¼ ePN

j¼1rjzjðxÞId‘2 , then TðxÞ is clearly a homeomorphism in ‘2 and the inverse operator is well defined by

T�1ðxÞ ¼ e�PN

j¼1rjzjðxÞId‘2 . It is easy to verify that kT�1ðhtxÞk has sub-exponential growth as t ! �1 for x 2 X. Hence

kT�1k is tempered. Since the mapping of h on �X has the same properties as the original one if we choose the trace r-algebrawith respect to �X to be denoted also by F , we can change our metric dynamical system with respect to �X, and still denotedthe symbols by ðX;F ;P; ðhtÞt2RÞ.

Denote .ðhtxÞ ¼PN

j¼1rjzjðhtxÞ, and consider the change in variable

vðtÞ ¼ T�1ðhtxÞuðtÞ ¼ e�.ðhtxÞuðtÞ;

where u is the solution of (3.2), then we get the evolution equation with random coefficients but without white noise

dvdtþ¼ �mAv � ðk� .ðhtxÞÞv � e�.ðhtxÞf ðe.ðhtxÞvÞ þ e�.ðhtxÞg ð3:5Þ

and initial condition vð0Þ ¼ v0 2 ‘2 ¼ H, where dvdtþ

is right-hand derivative of vðtÞ at t. For this type differential equation, wehave the following version of the Gronwall lemma (see [32]):

Lemma 3.2. Let xðtÞ be positive and satisfy the differential inequality

dxðtÞdtþ

6 gðtÞxðtÞ þ hðtÞ;

A. Gu, W. Ai / Commun Nonlinear Sci Numer Simulat 19 (2014) 1433–1441 1437

then

xðtÞ 6 xð0ÞeR t

0gðsÞds þ

Z t

0eR t

sgðrÞdrhðsÞds:

Now, we have the following result.

Theorem 3.3. Let T > 0 and v0 2 H be fixed, then the following statements hold:

(1) For every x 2 X, Eq. (3.5) has a unique solution vð�;x;v0Þ 2 Cð½0; TÞ; ‘2Þ in the sense of Carathéodory.(2) For each x 2 X, the mapping v0 2 ‘2 # vð�;x;v0Þ 2 Cð½0; TÞ; ‘2Þ is continuous, which implies the solution v of (3.5) contin-uously depends on the initial data v0.

Proof. (1) For any fixed T > 0 and v0 2 H and let v1;v2 2 Y , where Y is a bounded set in ‘2, then we have

kFðt;v1Þ � Fðt;v2Þk ¼ ke�.ðhtxÞf ðe.ðhtxÞv1Þ � e�.ðhtxÞf ðe.ðhtxÞv2Þk 6 CYkv1 � v2k;

where CY is a constant only depending on Y. This implies that the mapping Fðt;vÞ ¼ e�.ðhtxÞf ðe.ðhtxÞvÞ is locally Lipschitz withrespect to v and the Lipschitz constant is uniformly bounded in ½0; T�. By the standard arguments, we know that (3.5) pos-sesses a local solution vð�;x;v0Þ 2 Cð½0; TmaxÞ; ‘2Þ, where ½0; TmaxÞ is the maximal interval of existence of the solution of (3.5).Next, we need to show that the local solution is a global one. From (3.5) it yields that

ddtþkvðtÞk2 ¼ 2ð�mðAv ;vÞ � kkvk2 � ðe�.ðhtxÞf ðe.ðhtxÞvÞ;vÞ þ .ðhtxÞkvk2Þ þ 2ðe�.ðhtxÞg;vÞ

6 ð�kþ 2.ðhtxÞÞkvk2 þ 1k

e�2.ðhtxÞkgk2: ð3:6Þ

By Lemma 3.2, we get

kvðtÞk26 e�ktþ2

R t

0.ðhsxÞdskv0k2 þ kgk

2

ke�ktþ2

R t

0.ðhsxÞds

Z t

0e�2.ðhsxÞþks�2

R s

0.ðhrxÞdrds:

Denote

aðxÞ ¼ 2Z T

0j.ðhsxÞjds

and

bðxÞ ¼maxt2½0;T�

kgk2

ke�ktþ2

R t

0.ðhsxÞds

Z t

0e�2.ðhsxÞþks�2

R s

0.ðhrxÞdrds

!:

From Lemma 3.1, we know that aðxÞ; bðxÞ are well-defined. Then we have

kvðtÞk26 kv0k2eaðxÞ þ bðxÞ; ð3:7Þ

which implies that the solution v is defined in any interval ½0; T�.(2) Let u0;v0 2 H, and UðtÞ :¼ vðt;x;u0Þ;WðtÞ :¼ vðt;x;v0Þ be two solutions of (3.5). Then denoting ZðtÞ ¼ UðtÞ �WðtÞ,

we have

dZðtÞdtþ

¼ �mAZðtÞ � ðk� .ðhsxÞÞZðtÞ � e�.ðhsxÞðf ðe.ðhtxÞUðtÞÞ � f ðe.ðhtxÞWðtÞÞÞ

and

ddtþkZðtÞk2 ¼ 2ð�mðAZ; ZÞ � kkZk2 � e�.ðhsxÞðf ðe.ðhtxÞUðtÞÞ � f ðe.ðhtxÞWðtÞÞÞ; ZÞ þ 2.ðhtxÞkZk2

6 2e�.ðhsxÞkf ðe.ðhtxÞUðtÞÞ � f ðe.ðhtxÞWðtÞÞkkZk þ 2.ðhtxÞkZk26 2ðLþ .ðhtxÞÞkZk2

6 dkZk2;

where d ¼ 2ðLþmaxt2½0;T�j.ðhtxÞjÞ is well-defined, and L denotes the Lipschitz constant of f corresponding to a bounded setwhere U and W belong to. By Lemma 3.2 again, we obtain

kZðtÞk26 edtkZð0Þk2

and consequently

supt2½0;T�kUðtÞ �WðtÞk2

6 edTku0 � v0k2:

1438 A. Gu, W. Ai / Commun Nonlinear Sci Numer Simulat 19 (2014) 1433–1441

If u0 ¼ v0, then the above inequality indicates that the uniqueness and continuous dependence on the initial data of the solu-tions of (3.5). So the results hold, which completes the proof. h

Theorem 3.4. Eq. (3.5) generates a continuous RDS ðuðtÞÞtP0 over ðX;F ;P; ðhtÞt2RÞ, where

uðt;x;v0Þ ¼ vðt;x;v0Þ for v0 2 H; t P 0 and for all x 2 X:

Moreover,

wðt;x;v0Þ ¼ TðhtxÞwðt;x; T�1ðxÞv0Þ for v0 2 H; t P 0 and for all x 2 X;

then w is another RDS for which the process ðx; tÞ ! ðwðt;x;v0ÞÞ solves (3.2) for any initial condition v0 2 H.

Proof. The continuity of u is due to Theorem 3.3. The measurability of w follows from the properties of T. Here, we onlyremain to prove the conjugacy between u and w. By using the chain rule, we have

dðTuÞ ¼ TduþXN

j¼1

rjTu}dzj

¼ Tð�mAu� ðk� .Þu� T�1f ðTuÞ þ T�1gÞdt � .Tudt þXN

j¼1

rjTu}dLj

¼ ð�mAðTuÞ � kTu� f ðTuÞ þ gÞdt þXN

j¼1

rjTu}dLj;

that is

dðTuÞdtþ

¼ �mAðTuÞ � kTu� f ðTuÞ þ g þXN

j¼1

rjTu} dLj

dt;

which completes the proof. h

4. Existence of global random attractors

In this section, we will prove the existence of a global random attractor for the SLDS generated by Eq. (1.1). Since the ran-dom dynamical systems u and w are conjugated, we only have to consider the RDS u. Firstly, we have our main result

Theorem 4.1. The SLDS u generated by Eq. (3.5) has a unique global random attractor.In order to prove Theorem 4.1, we will use Proposition 2.6. We first need to prove there exists an absorbing set for u in

DðHÞ. Next, we will show the RDS u is random asymptotically null in the sense of Definition 2.5.

Lemma 4.2. There exists a closed random tempered set KðxÞ 2 DðHÞ such that for all B 2 DðHÞ and a.e. x 2 X there existstBðxÞ > 0 such that

uðt; h�txÞBðh�txÞ � KðxÞ for all t P tBðxÞ:

Proof. Let us start with vðtÞ ¼ uðt;x;v0Þ. Then by (3.6), we have

kvðtÞk26 e�ktþ2

R t

0.ðhsxÞdskv0k2 þ kgk

2

ke�ktþ2

R t

0.ðhsxÞds

Z t

0e�2.ðhsxÞþks�2

R s

0.ðhrxÞdrds:

Now, by replacing x with h�tx and v0 with e�.ðh�txÞv0, respectively, in the expression u, we obtain

kuðt; h�tx; e�.ðh�txÞv0Þk26 e�ktþ2

R t

0.ðhs�txÞdske�.ðh�txÞv0k2 þ kgk

2

ke�ktþ2

R t

0.ðhs�txÞds

Z t

0e�2.ðhs�txÞþks�2

R s

0.ðhr�txÞdrds

6 e�kt�2.ðh�txÞþ2R t

0.ðhs�txÞdskv0k2 þ kgk

2

k

Z t

0e�2.ðhs�txÞþkðs�tÞ�2

R t

s.ðhr�txÞdrds 6 e�kt�2.ðh�txÞþ2

R 0

�t.ðhsxÞdskv0k2

þ kgk2

k

Z 0

�te�2.ðhsxÞþks�2

R 0

s.ðhrxÞdrds 6 e�kt�2.ðh�txÞþ2

R 0

�t.ðhsxÞdskv0k2 þ kgk

2

k

Z 0

�1e�2.ðhsxÞþks�2

R 0

s.ðhrxÞdrds: ð4:1Þ

Due to Lemma 3.1, we know that

A. Gu, W. Ai / Commun Nonlinear Sci Numer Simulat 19 (2014) 1433–1441 1439

Z 0

�1e�2.ðhsxÞþks�2

R 0

s.ðhrxÞdrds < þ1:

Considering for any v0 2 Bðh�txÞ, we have

kuðt; h�tx; e�.ðh�txÞv0Þk26 e�kt�2.ðh�txÞþ2

R 0

�t.ðhsxÞdsdðBðh�txÞÞ2 þ

kgk2

k

Z 0

�1e�2.ðhsxÞþks�2

R 0

s.ðhrxÞdrds:

Denoting

R2ðxÞ ¼ 2kgk2

k

Z 0

�1e�2.ðhsxÞþks�2

R 0

s.ðhrxÞdrds ð4:2Þ

and noting that

limt!þ1

e�kt�2.ðh�txÞþ2R 0

�t.ðhsxÞdsdðBðh�txÞÞ2 ¼ 0;

we conclude that

KðxÞ ¼ BHð0;RðxÞÞ ð4:3Þ

is an absorbing closed random set. It remains to show that KðxÞ 2 DðHÞ. Indeed, from Definition 2.2 (iv), for all c > 0, we get

e�ctR2ðh�txÞ ¼ 2e�ct kgk2

k

Z 0

�1e�2.ðhs�txÞþks�2

R 0

s.ðhr�txÞdrds ¼ 2e�ct kgk

2

k

Z �t

�1e�2.ðhsxÞþkðsþtÞ�2

R 0

s.ðhrxÞdrds! 0 as t !1:

Thus, the proof is complete. h

Lemma 4.3. Let v0ðxÞ 2 KðxÞ be the absorbing set given by (4.3). Then for every � > 0, there exist eT ð�;x;KðxÞÞ > 0 andeNð�;x;KðxÞÞ > 0, such that the solution u of problem (3.5) is random asymptotically null, that is

supv2KðxÞ

Xjij>eNð�;x;KðxÞÞjuiðt; h�tx;vðh�txÞj2 6 �2; 8t P eT ð�;x;KðxÞÞ:

Proof. Choose a smooth cut-off function satisfying 0 6 qðsÞ 6 1 for s 2 Rþ and qðsÞ ¼ 0 for 0 6 s 6 1, qðsÞ ¼ 1 for s P 2. Sup-pose there exists a constant c0 such that jq0ðsÞj 6 c0 for s 2 Rþ.

Let N be a fixed integer which will be specified later, and set x ¼ ðq jijN

� �v iÞ

i2Z. Then taking the inner product of (3.5) with x

in H, we obtain

ddtþ

Xi2Z

qjijN

� �juij

2 ¼ �2mðAu; xÞ � 2ðk� .ðhtxÞÞXi2Z

qjijN

� �juij

2 � 2e�.ðhtxÞXi2Z

qjijN

� �fiðe.ðhtxÞuiÞui

þ 2e�.ðhtxÞXi2Z

qjijN

� �giui: ð4:4Þ

We now estimate terms in (4.4) one by one. First, we have

ðAu; xÞ ¼ ðeBu; eBxÞP �2c0

Nkuk2

: ð4:5Þ

For the second term in (4.4), it follows from the assumptions on the nonlinearity fi that

�1 < �2e�.ðhtxÞXi2Z

qjijN

� �fiðe.ðhtxÞuiÞui 6 0:

For the last term in (4.4),

Xi2Z

qjijN

� �giui ¼

XjijPN

qjijN

� �giui 6

k2

XjijPN

qjijN

� �juij

2 þ 12k

XjijPN

jgij2: ð4:6Þ

Combining (4.4)–(4.6), we get

ddtþ

Xi2Z

qjijN

� �juij

2 þ ðk� 2.ðhtxÞÞXi2Z

qjijN

� �juij

26

4mc0

Nkuðt;x; e�.ðxÞv0Þk2 þ 1

ke�.ðhtxÞ

XjijPN

jgij2:

By using Lemma 3.2, for t P TK ¼ TKðxÞ, it follows that

1440 A. Gu, W. Ai / Commun Nonlinear Sci Numer Simulat 19 (2014) 1433–1441

Xi2Z

qjijN

� �juiðt;x; e�.ðxÞv0ðxÞÞj2 6 e

�kðt�TKÞþ2R t

TK.ðhsxÞdsX

i2ZqjijN

� �juiðt;x; e�.ðxÞv0ðxÞÞj2 ð4:7Þ

þ4mc0

N

Z t

TK

e�kðt�sÞþ2R t

s.ðhsxÞdskuðs;x; e�.ðxÞv0ðxÞÞk2ds ð4:8Þ

þ1k

XjijPN

jgij2Z t

TK

e�kðt�sÞþ2R t

s.ðhsxÞds�.ðhtxÞds: ð4:9Þ

Now, substitute h�tx for x and estimate each term from (4.7)–(4.9). In (4.1), with t replaced with TK and x with h�tx,respectively, it follows from (4.7) that

e�kðt�TKÞþ2

R t

TK.ðhs�txÞdsX

i2ZqjijN

� �juiðTK; h�tx; e�.ðh�txÞv0ðh�txÞÞj2

6 e�kt�2.ðh�txÞþ2R t

0.ðhsxÞdskv0k2 þ kgk

2

k

Z TK

0e�2.ðhs�txÞþkðs�tÞþ2

R t

s.ðhr�txÞdrds

6 e�kt�2.ðh�txÞþ2R t

0.ðhsxÞdskv0k2 þ kgk

2

k

Z TK�t

�te�2.ðhsxÞþks�2

R 0

s.ðhrxÞdrds: ð4:10Þ

By Lemma 3.1, there exists a T1ð�;x;KðxÞÞ > TKðxÞ, such that if t > T1ð�;x;KðxÞÞ, then

e�kðt�TKÞþ2

R t

TK.ðhs�txÞdsX

i2ZqjijN

� �juiðTK; h�tx; e�.ðh�txÞv0ðh�txÞÞj2 6

�2

3: ð4:11Þ

Next, from (4.1) and (4.8), it follows that

4mc0

N

Z t

TK

e�kðt�sÞþ2R t

s.ðhs�txÞdskuðs; h�tx; e�.ðh�txÞv0ðh�txÞÞk2ds

64mc0

Nkv0k2ðt � TKÞe�kt�2.ðh�txÞþ2

R t

0.ðhs�txÞds þ 4mc0kgk2

kN

Z t

TK

Z s

0e�2.ðhs�txÞþkðs�tÞþ2

R t

s.ðhr�txÞdrdsds

64mc0

Nkv0k2ðt � TKÞe�kt�2.ðh�txÞþ2

R t

0.ðhs�txÞds þ 4mc0kgk2

kN

Z t

TK

Z s�t

�te�2.ðhsxÞþks�2

R 0

s.ðhrxÞdrdsds: ð4:12Þ

By Lemma 3.1, there exist T2ð�;x;KðxÞÞ > TKðxÞ and N1ð�;x;KðxÞÞ > 0 such that if t > T1ð�;x;KðxÞÞ andN > N1ð�;x;KðxÞÞ, then

4mc0

N

Z t

TK

e�kðt�sÞþ2R t

s.ðhs�txÞdskuðs; h�tx; e�.ðh�txÞv0ðh�txÞÞk2ds 6 �

2

3: ð4:13Þ

Since g 2 H, by using Lemma 3.1 again, we find that there exists N2ð�;x;KðxÞÞ > 0 such that if N > N2ð�;x;KðxÞÞ, then from(4.9),

1k

XjijPN

jgij2Z t

TK

e�kðt�sÞþ2R t

s.ðhsxÞds�.ðhtxÞds 6 �

2

3: ð4:14Þ

Let

eT ð�;x;KðxÞÞ ¼maxfT1ð�;x;KðxÞÞ; T2ð�;x;KðxÞÞg;eNð�;x;KðxÞÞ ¼maxfN1ð�;x;KðxÞÞ;N2ð�;x;KðxÞÞg:

Then from (4.11), (4.13) and (4.14), for t > eT ð�;x;KðxÞÞ and N > eNð�;x;KðxÞÞ, we get

XjijP2N

juiðt; h�tx; e�.ðh�txÞv0ðh�txÞÞj2

6

Xi2Z

qjijN

� �juiðt; h�tx; e�.ðh�txÞv0ðh�txÞÞj2 6 �2;

which implies the conclusion. h

We are now in a position to prove our main result.Proof of Theorem 4.1. The desired result follows directly from Lemmas 4.2 and 4.3 and Proposition 2.6.

A. Gu, W. Ai / Commun Nonlinear Sci Numer Simulat 19 (2014) 1433–1441 1441

Acknowledgment

The authors thank the anonymous referees for very helpful comments and suggestions which largely improve the presen-tation and quality of the manuscript.

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