Abstract
We study nonautonomous stochastic sine-Gordon lattice systems with random coupled coefficients and multiplicative white noise. We first consider the existence of random attractors in a weighted space for this system and then establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.
1. Introduction
Stochastic lattice differential equations arise naturally in a wide variety of applications where the spatial structure has a discrete character and uncertainties are taken into account. In recent years, some works have been done regarding the existence of random attractors for stochastic lattice differential equations (see, e.g., [1–11]). Of those works, Bates et al. [2] considered the existence of random attractors for first-order nonautonomous stochastic lattice system driven by multiplicative white noise. Zhou and Wei [11] considered the existence of random attractors for second-order lattice system with random coupled coefficients and multiplicative white noise.
Motivated by [2, 11], we will study the asymptotic behavior of solutions of the following nonautonomous stochastic sine-Gordon lattice systems with random coupled coefficients and multiplicative white noise: for every and ,with the initial data where , and are positive constants; ; ; ∘ denotes the Stratonovich sense of the stochastic term; are random variables; is a Brownian motion (Wiener process) on a probability space , where , is the Borel -algebra induced by the compact open topology of , and is the Wiener measure on (see [12]); and is a metric dynamical system defined on , where here for all . Then is an ergodic metric dynamical system.
In practice, the coupled mode between two nodes (say, adjacent nodes) is usually random. It is then of great importance to investigate stochastic lattice system with random coupled coefficients. To the best of our knowledge, there are no results on nonautonomous stochastic sine-Gordon lattice systems with random coupled coefficients and multiplicative white noise.
In this paper, we first consider the existence of a tempered random attractor in a weighted space for stochastic sine-Gordon lattice systems (1), which attracts the random tempered bounded sets in pullback sense. Then, we consider the dependence of attractors on the parameters of the system (1) and establish the upper semicontinuity of the random attractor as the intensity of noise approaches zero.
The rest of this paper is organized as follows. In the next section, we present some mathematical setting for system (1). In Section 3, we mainly consider the existence of a tempered random attractor in a weighted space of infinite sequences for system (1). Then in Section 4, we consider the upper semicontinuity of the tempered random attractor for system (1) as .
2. Mathematical Settings
We first introduce a weighted space of infinite sequences. Let be a positive-valued function from into which satisfieswhere and are positive constants. For example, for , [13] and satisfy condition (3), where . Define and with norm and inner product for . We write as if . Then is a separable Hilbert space with the norm .
For any , define an inner product on by then the norm induced by inner product is equivalent to the norm induced by the inner product . We write ; then and are both separable Hilbert spaces.
Note that system (1) can be rewritten aswhere , is a given time dependent sequence, and is a linear operator defined by
To convert problem (6) into a random differential equation, let , and , which is an Ornstein-Uhlenbeck process on and solve the Ornstein-Uhlenbeck equation , where for . From [12, 14, 15], it is known that the random variables are tempered, and there is a -invariant set of full measure such that is continuous in for every .
Let ; then (6) can be written as the following equivalent random system with random coefficients: for every and ,
We will consider (8) for and write as from now on. In order to obtain the existence and uniqueness of solutions to problem (8), we make the following assumptions on and the coefficients , for :(A1)Letting belongs to with respect to for each , and is tempered.(A2).(A3)Let , and There exists a positive constant such that
We call a mild solution of the following random lattice differential equations:where , if and
Theorem 1. Let and (A1)-(A2) hold. Then for every and and any initial data , problem (8) admits a unique mild solution with , being continuous in ; if . Moreover, (8) generates a continuous cocycle over and with state space : for , and ,
We can prove Theorem 1 by Theorem 6.1.7 in [16] and Definition 2.1 in [2]. We omit it here.
3. Existence of Random Attractors
We first provide some sufficient conditions for the existence of random attractors for nonautonomous RDSs in weighted spaces of infinite sequences in [2].
In the following, let be a separable Banach space and let be the collection of all tempered families of nonempty bounded subsets of .
Definition 2. Let be a continuous cocycle on over and .(1)A family is called a random absorbing set for if, for all and and for every , there exists such that(2) A family is called a random attractor for if for all , , and , (i) is compact in and is measurable in with respect to ; (ii) is invariant, that is, ; and (iii) for every , where is the Hausdorff semidistance given by , for any .
Theorem 3. Let be a continuous cocycle on over and . Suppose the following.(a)There exists a bounded closed random absorbing set such that, for any , , and , there exists yielding ;.(b)For each , and for any , there exist and such thatThen possesses a unique random attractor in given by, for every and ,
Next, we will use Theorem 3 to prove the existence of a random attractor for the continuous cocycle in under conditions (A1)–(A3).
Theorem 4. If (A1)–(A3) hold, then, for every , , and and for any , there exists such that, for all and , the solution of (8) satisfieswhere
Proof. For each , there exists a sequence of continuous functions in (see [17]) such thatand for . Consider the following random differential equations:where It is easy to see that (23) has a unique mild solution satisfying (23). Taking the inner product of the first equation in (23) with in and the second equation with in , then we have thatSumming the two equations of (24), we find thatNote thatBy (3), we getwhere
From (25)–(28), we obtain that, for ,Recalling that and in (10), then we havewhere is as in (21). Then we obtain from (32) that, for ,From (33) and by replacing by , we haveNote that (34) holds with and being replaced by and ; then we haveLetthen . By (9), we find that there exists such that, for , from which, along with , we haveTherefore, there exists such that, for all ,Since and are tempered, then is tempered. Then by (A3), we can verify that the following integrals are convergent:Thus the theorem follows from (35), (39), and (40).
Theorem 5. Assume that (A1)–(A3) hold. Then the continuous cocycle associated with (8) has a unique random attractor .
Proof. By Theorem 3, it suffices to prove that, for every , , , and and for any , there exist and such that, for all and , the solution of (8) satisfiesChoose a smooth increasing function such thatand there exists a positive constant such that for .
Let be a mild solution of (8) with . For any given define , by if and otherwise.
For any , let , be the solution of (23).
Then taking the inner product of the first equation in (23) with in and the second equation with in , we have thatFor each term of (43), it has been checked thatPutting above inequalities into (43), we obtainwhere is as in (21). Multiplying (45) by and then integrating over with , we obtainReplacing and in (46) by and , respectively, and letting , then we have thatWe now estimate each term on the right-hand side of (47). For the first term on the right-hand side of (47), since and is tempered, then there exists such that if , thenFor the second term on the right-hand side of (47), by (A3), there exist and , such that if and , thenFor the third term on the right-hand side of (47), by (A1) and Theorem 4, there exist and such that, for all and ,Let and . Then it follows from (48)–(50) that, for all and , we obtainfor any . Let ; we have that (41) holds. This completes the proof.
4. Upper Semicontinuity of Random Attractors
First, we present a criteria concerning upper semicontinuity of nonautonomous random attractors with respect to a parameter in [18].
Theorem 6. Let be a continuous cocycle on over and . Suppose the following.
(i) has a closed measurable random absorbing set in and a unique random attractor in .
(ii) For each and , andwhere is a positive-valued tempered random variable.
(iii) There exists such that, for every and , is precompact in .
(iv) For , , , when , and with when , it holds:Then for and ,
Next, we will use Theorem 6 to consider an upper semicontinuity of random attractors to as .
Theorem 7. Assume that (A1)–(A3) hold. Then for every and , one has
Proof. Let be as in Theorem 4.
(i) By Theorems 4 and 5, has a closed measurable random absorbing set , where , and a unique random attractor in , for each and ,
(ii) Given . By (20), we haveSo, for every and , Moreover, is a closed tempered random absorbing set for the continuous cocycle associated with the limiting system:(iii) Given . Let us prove the precompactness of for every and . For one thing, by Theorem 5, for every , , , and , there exist and such that, for all and , the solution of (8) satisfiesfrom which, along with (59) and the invariance of , we have, for every , , and ,For another, by (59) we find that the set is bounded in a finite-dimensional space and hence is precompact in .
(iv) Let and be mild solutions of (8) and (58) with initial data and , respectively, and set , , .
Let and be the solutions of the following random differential equations with initial data:respectively. Then, , and satisfy the differential equations (62) and (63), respectively. Moreover, and are limit functions of subsequences of and . So is a limit function of a subsequence of in , and satisfiesNow let us estimate . Taking the inner product of (64) with and in and , respectively, then we getWe now estimate the terms in (65):Then it follows from (66) thatwhere , is as in (10) and . Applying Gronwall’s inequality to (67) from to , we haveWe now replace in the above by to yield From (69), we see that, for , , , , and , with ,Let be a sequence of numbers with when . Then and being limit functions of subsequences of and in imply that, for , , , , and , with ; the following holds:The proof is completed.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
The authors would like to express their sincere thanks to the anonymous referees for their time and comments. This work is supported by the National Natural Science Foundation of China under Grant nos. 11326114, 11401244, and 11471290; Natural Science Research Project of Ordinary Universities in Jiangsu Province under Grant no. 14KJB110003; Zhejiang Normal University Foundation under Grant no. ZC304011068; and Zhejiang Natural Science Foundation under Grant no. LY14A010012.