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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 279509, 23 pages
http://dx.doi.org/10.1155/2013/279509
Research Article

-Random Attractors for Stochastic Reaction-Diffusion Equation on Unbounded Domains

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

Received 11 May 2013; Accepted 30 September 2013

Academic Editor: Grzegorz Lukaszewicz

Copyright © 2013 Gang Wang and Yanbin Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the random dynamical system generated by a stochastic reaction-diffusion equation with additive noise on the whole space and prove the existence of an -random attractor for such a random dynamical system. The nonlinearity is supposed to satisfy the growth of arbitrary order (). The -asymptotic compactness of the random dynamical system is obtained by using an extended version of the tail estimate method introduced by Wang (1999) and the cut-off technique.

1. Introduction

In this paper, we consider the asymptotic behavior of solutions to the following stochastic reaction-diffusion equation (SRDE) with additive noise in the entire space : with the initial condition where is a positive constant, is a given function in , for each , for some ,   are independent two-sided real-valued Wiener processes on a probability space which will be specified below, and is a nonlinear function satisfying the following conditions (see, e.g., [1, 2]). For all and , where and are positive constants, , , and .

As we know, the asymptotic behavior of a random dynamical system (RDS) is characterized by random attractors, which were first introduced by Crauel and Flandoli [3] and Schmalfuss [4] and then developed in [1, 2, 512] and among others. Recently, the existence of random attractors of the RDS associated with problem (1)-(2) was studied by many authors. For example, in [1, 2] the authors proved the existence of -random attractor and -random attractor, respectively, in the case of additive noise. Wang and Zhou obtained -random attractor in [12] and Li et al. proved the existence of -random attractor in bounded domains in [10] in the case of multiplicative noise. A necessary and sufficient condition for the existence of random attractors for the so-called quasicontinuous RDS was established in [9], and in the most recent papers [13, 14], the author employed this result to prove the existence of random attractors for some reaction-diffusion equations with additive noise and multiplicative noise on , respectively, when the domain is bounded. In this paper, we study the existence of -random attractor with additive noise for the same problem in the entire space .

For our problem, there are two difficulties when we consider the existence of -random attractor. The first is the lack of compactness of Sobolev embeddings when the domain is unbounded. It is worth mentioning that in deterministic case differential equations of this type were extensively studied in both autonomous and nonautonomous cases and in both bounded domains and unbounded domains [1529]. In the case of unbounded domains the difficulty of noncompact embeddings can be overcome by the energy equation approach introduced by Ball in [30, 31] and other methods. We are interested in the method used in [22] for the deterministic version of the initial problem (1)-(2) on . In [22] the author approached by a bounded ball and found that the approximation error of the norm of solutions is arbitrary small uniformly for large time, and thus they proved asymptotic compactness by passing the limit of the energy equation. More recently, the idea of the tail estimate was used in [1] to prove the existence of random attractor in for the SRDE (1)-(2). In this paper, we use an extended version of the tail estimate described in [22] to overcome the difficulty of noncompact embeddings.

Another difficulty is that one can not differentiate the stochastic equation with respect to time in usual sense. In the case of deterministic equation, by differentiating the reaction-diffusion equation with respect to , one can prove the existence of or ( is bounded) attractors; see [24, 27, 29, 32] for autonomous equations and [23, 26, 33] for non-autonomous equations. But in stochastic case this idea breaks down, since, as we know, neither the Winner process nor the Ornstein-Uhlenbeck process is differentiable with respect to in usual sense. However, this is only a matter of method or estimate. In [25], the author used a result for compactness in introduced in [17] to establish the asymptotic compactness in without differentiating the equation. Unfortunately, the growth order is restricted in that case. In this paper, we overcome this drawback by using an appropriate estimate motivated by the works in [19] and the estimate is accurate enough so that we needn't differentiate the equation as usual.

This paper is organized as follows. In Section 2, we recall some basic notions of bispaces random attractors for RDS. In Section 3, we transform the problem (1)-(2) into a parameterized evolution equation and obtain the corresponding RDS. In Section 4, we give some uniform estimates of the solutions as . In Section 5, we prove the asymptotic compactness and the existence of an -random attractor.

Throughout this paper, we denote by the norm of Banach space and by the inner product in Hilbert space . The inner product and norm of are written as and , respectively. We also use to denote the norm of (, ) and to denote the modular of . The letter denotes any positive constant which may be different from line to line or even in the same line (sometimes for special case, we also denote the different positive constants by ).

2. Preliminaries and Abstract Results

In this section, we first recall some basic concepts related to random attractors for RDS (see [1, 3, 58, 34] for more detail) and then give some abstract results on the existence of -random attractors.

2.1. Preliminaries

Let , be two Banach spaces with Borel -algebra and , respectively, and let be a probability space.

Definition 1. is called a metric dynamical system (MDS) if is -measurable, and is the identity on , = for all , and for all .

Definition 2. An RDS on over an MDS is a mapping , which is -measurable and satisfies that, for -a.e. ,(i) = id on ;(ii) (cocycle property) on for all .
An RDS is said to be continuous on if is continuous for all and -a.e. .

Definition 3. A random set is a set-valued map , , which satisfies that, for each , the map is measurable. A random set is called a random closed (compact) set if is closed (compact) for all . A random set is called a random bounded set if there exist and a random variable such that, for all ,

Definition 4. A random bounded set of is called tempered with respect to if, for -a.e. , where
A random variable is called tempered with respect to if, for -a.e. ,
Next, we introduce some notions about the bi-spaces random attractors which are motivated by the works in [2, 20, 25, 35]. We assume that is an RDS on and over an MDS , respectively. Let denote the family of all nonempty subsets of and the class of all families . , and can be defined in the same way. We consider the given nonempty subclasses , where , .

Definition 5. A family is said to be -random absorbing for if, for every , there exists such that, for -a.e. ,

Definition 6. A family is said to be -random attracting for if, for every ,  we have, for -a.e. , where denotes the Hausdorff semi-distance between and in ; that is,

Definition 7. The RDS is said to be -asymptotically compact if, for -a.e. , has a convergent subsequence in whenever and with .

Definition 8. random set is said to be an -random attractor if the following conditions are satisfied for -a.e. , (i) is closed in and compact in ;(ii) is invariant; that is, for all ;(iii) attracts every random set in in the norm topology of in the sense of (12).

2.2. Abstract Results

Now, we present the main abstract results. Recall that a collection of random subsets is called inclusion closed if whenever is an arbitrary random set and is in with for all , then .

The following theorem is an adaptation of a result of [25] to the case of RDS. The proof is similar to that of [25], and here we omit it.

Theorem 9. Let be a continuous RDS on and an RDS on over , respectively, and and are inclusion closed.(i) Case  1  ()  (see [1]). Assume that the family is a closed -random absorbing set for and is -asymptotically compact. Then has a unique -random attractor which is given by where denotes the closure of with respect to the norm topology in .(ii)  Case  2  (). If the assumptions in (i) are satisfied, moreover, we assume that is -random absorbing and is -asymptotically compact. Then has an -random attractor which is given by where is the -random absorbing set in (i).

In the following of this paper we only consider , , and , , where and denote the collections of all tempered random subsets of and , respectively.

Theorem 10. Assume that is an RDS on and , respectively, and then is -asymptotically compact if(i) for every , -a.e. and every , there exist and such that, for all , (ii) is -asymptotically compact, ,where , , , and is the identical function on .

Proof. It suffices to check that, for all and -a.e. , we can extract a Cauchy subsequence from , whenever and . We assume that there is of full -measure such that assumption (i) holds for every . We now fix and , and then by (i) there exist and such that for all , On the other hand, by (ii), is -asymptotically compact, for all . For the above , there is a subsequence such that is convergent in . Therefore, there exists an integer such that for all , we have The proof is complete.

Remark 11. If we replace by other Banach spaces in Theorem 10, such as , and , the corresponding results also hold true. In particular, in the deterministic case, it is the exact method used in [22] when is replaced by .

3. The Reaction-Diffusion Equation on with Additive Noise

We consider the probability space where is the Borel -algebra induced by the compact-open topology of , and the corresponding Wiener measure on . Then we will identify with Define the time shift by and then is an MDS.

We now translate the stochastic equation (1)-(2) into a deterministic equation with a random parameter.

To this end, we consider the one-dimensional Ornstein-Uhlenbeck process given by which solves the Itô differential equation Note that the random variable is tempered and is -a.e. continuous in . Therefore, it follows from the Proposition in [34] that there exists a tempered function such that where satisfies that for -a.e. , Therefore, for -a.e. , Putting then by (23) we have

Remark 12. From (24) and (27), we can easily show that the sum is bounded by with a deterministic positive constant . In the following of this paper, we use the symbols and to denote the random variables in (24).

In order to show that the initial problem (1)-(2) generates an RDS, we set . Then we can consider the following evolution equation with random parameter but without white noise: with initial value condition

From [1, 2], we see that for -a.e. and all , the parameterized evolution equation (30)-(31) with conditions (3)–(6) has a unique solution for every . Furthermore, is continuous with respect to in , for all .

As the process is the solution to the problem (1)-(2) in a certain sense. We now define a mapping by for all . Then is a continuous RDS on and an RDS on respectively associated with the initial value problem of SRDE (1)-(2) on .

Theorem 13 (see [1, 2]). Assume that and (3)–(6) hold. Then the RDS generated by (1)-(2) has a unique -random attractor and has a unique -random attractor ; furthermore, we have .

4. Uniform Estimates of Solutions

4.1. -Random Absorbing Set and Some Useful Estimates

The next lemma shows that has a tempered -random absorbing set.

Lemma 14 (see [1]). Assume that and (3)–(6) hold. Let and . Then, for -a.e. , there exists such that, for all , where is a constant independent of , , and .

We now give some new estimates for the solution of (30)-(31).

Lemma 15. Assume that and (3)–(6) hold. Let and . Then, for -a.e. , there exists , such that the solution of (30)-(31) satisfies that, for all and for all , where is a constant independent of , , and .

Proof. The first assertion was proved in [2] in the case of , and the case for can be obtained by slightly modifying the proof of Lemma 4.4 in [2] (in fact, for , (36) also holds true, and we will use this result in (55)), and here we omit it.
Now, we prove the second assertion. Multiplying (30) with and integrating over , we get Since , we have For the nonlinearity, similar to and in [2], we have From (39)-(40) and (42)-(43), we get
On the other hand, multiplying (30) by and integrating over , we get the results in [1]: By Hölder inequality, , we can convert (45) into For any , integrating (47) over and using (46), we get Next, fix and integrate (44) over to get Integrating the above inequality with respect to over and using (48), we obtain, for all , Replacing by first, then substituting for in the aforementioned inequality, and noting that , we have where we have used (26) in the pervious inequality. Noting that with and , we get from (51) that there exists such that, for all and for all , That is, (37) holds true.
To prove (38), we take the inner product of (30) with in , and using in [1], we get This implies that Integrating (47) over and substituting for , then from (36), we get Obviously, from (51) we can easily see that (37) also holds for ; then, by (37), (47), (54)-(55), and a similar procedure as the proof of (37), one can show that, for all , and for all , The proof is complete.

Lemma 16. Assume that and (3)–(6) hold. Let and . Then, for -a.e. , there exists , such that the solution of (30)-(31) satisfies that for all ,

Proof. Integrating (44) over , we get Replacing by in the aforementioned inequality, it yields that Lemma 15 and the aforementioned inequality imply that there exists , , such that (57) holds.
Next, taking the inner product of (30) with in , and using (4), we obtain that is, We now integrate (62) over to obtain Replacing by , we get Equations (36), (38) and (57) together imply that (58) is also true. The proof is complete.

4.2. Tail Estimate in

We next estimate “the tail” of the solution to the problem (1)-(2) in .

Lemma 17. Assume that and (3)–(6) hold. Let and . Then, for -a.e. and for every , there exist and , such that the solution of (30)-(31) satisfies that ,

Proof. By Lemma 4.6 in [1], it suffices to prove that Let be a smooth function defined on such that , for all , and Then there is a positive constant such that for all .
Multiplying (30) by and integrating with respect to over , we get The second term of the left-hand side is bounded by Similarly, the forth term of the left-hand side of (68) is bounded by For the last term of the right-hand side of (68), we have We next consider the nonlinear term in (68). Since
We now estimate each term in the right-hand side of (72). Using (4), the property of , and Cauchy's inequality, we see that the first term of the right-hand side of (72) is bounded by By (6), we can estimate the second term of the right-hand side of (72) as follows: For the third term of the right-hand side of (72), by using (5), we have For the last term of the right-hand side of (72), by using (4) and Young's inequality, we find Putting (73)–(76) together into (72), it yields that Then by (68)–(71) and (77), we get In particular, Let and integrate the aforementioned inequality from to : <