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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 670786, 22 pages
Research Article

Global Mild Solutions and Attractors for Stochastic Viscous Cahn-Hilliard Equation

1Department of Mechanic, Mechanical College, Tianjin University, Tianjin 300072, China
2Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China
3Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

Received 22 March 2011; Accepted 19 May 2011

Academic Editor: Nicholas D. Alikakos

Copyright © 2011 Xuewei Ju et al. 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.


This paper is devoted to the study of mild solutions for the initial and boundary value problem of stochastic viscous Cahn-Hilliard equation driven by white noise. Under reasonable assumptions we first prove the existence and uniqueness result. Then, we show that the existence of a stochastic global attractor which pullback attracts each bounded set in appropriate phase spaces.

1. Introduction

This paper is devoted to the existence of mild solutions and global asymptotic behavior for the following stochastic viscous Cahn-Hilliard equation: Δ𝑑((1𝛼)𝑢𝛼Δ𝑢)+2𝑡𝑢Δ𝑓(𝑢)𝑑𝑡=𝑑𝑊,(𝑥,𝑡)𝐺×0,,(1.1) subjected to homogeneous Dirichlet boundary conditions 𝑢𝑡(𝑥,𝑡)=0,(𝑥,𝑡)𝜕𝐺×0,,(1.2) in dimension 𝑛=1,2 or 3, where 𝐺=𝑛𝑖=1(0,𝐿𝑖) in 𝑅𝑛, and 𝛼[0,1] is a parameter, 𝑓 is a polynomial of odd degree with a positive leading coefficient 𝑓(𝑥)=2𝑝1𝑘=1𝑎𝑘𝑥𝑘,𝑎2𝑝1>0.(1.3)

In deterministic case, the model was first introduced by Novick-Cohen [1] to describe the dynamics of viscous first order phase transitions, which has been extensively studied in the past decades. The existence of global solutions and attractors are well known; moreover, the global attractor 𝒜𝛼 of the system has the same finite Hausdorff dimension for different parameter values 𝛼. One can also show that 𝒜𝛼 is continuous as 𝛼 varies in [0,1]. See [2] for details and [1] for recent development.

While the deterministic model captures more intrinsic nature of phase transitions in binary, it ignores some random effects such as thermal fluctuations which are present in any material. In recent years, there appeared many interesting works on stochastic Cahn-Hilliard equations. Cardon-Weber [3] proved the existence of solution as well as its density for a class of stochastic Cahn-Hilliard equations with additive noise using an appropriate convolution semigroup (in the sense of that in [4]) posed on cubic domains. The authors in [5] derived the existence for a generalized stochastic Cahn-Hilliard equation in general convex or Lipschitz domains. The main novelty was the derivation of space-time Hölder estimates for the Greens kernel of the stochastic problem, by using the domains geometry, which can be very useful in many other circumstances. In [6], the asymptotic behavior for a generalized Cahn-Hilliard equation was studied, which can also act as a very good toy model for treating the stochastic case.

Instead of deterministic viscous Cahn-hilliard equation, here, we consider the general stochastic equation (1.1) which is affected by a space-time white noise. In such a case, new difficulties appear, and the resulting stochastic model must be treated in a different way. Fortunately, the rapidly growing theory of random dynamical systems provides an appropriate tool. Crauel and Flandoli [7] (see also Schmalfuss [8]) introduced the concept of a random attractor as a proper generalization of the corresponding deterministic global attractor which turns out to be very helpful in the understanding of the long-time dynamics for stochastic differential equations. In this present work, we first establish some existence results on mild solutions. Then, by applying the abstract theory on stochastic attractors mentioned above, we show that the system has global attractors in appropriate phase spaces.

In case 𝛼=0, (1.1) reduces to the stochastic Cahn-Hilliard equation which was studied in [9], where the authors obtain the existence and uniqueness of the weak solutions to the initial and Neumann boundary value problem in some phase spaces under appropriate assumptions on noise. Here, we make slightly stronger assumptions on noise and prove existence and uniqueness of mild solutions with higher regularity. Furthermore, we show the existence of random attractors in appropriate phase spaces.

This paper is organized as follows. In Section 2, we first make some preliminary works, then we state our main results. In Section 3, we consider the solutions of the the linear part of the system (1.1)-(1.2) and stochastic convolution. Regularities of solutions will also be addressed in this part. Section 4 consists of some investigations on the Stochastic Lyapunov functional of the system. The proofs on the existence results for mild solutions and global attractors will be given in Sections 5 and 6, respectively. Finally, the last section stands as an appendix for some basic knowledge of random dynamical system(RDS).

2. Preliminaries and Main Results

In this section, we first make some preliminary works, then we state explicitly our main results.

2.1. Functional Spaces

Let (,) and || denote respectively the inner product and norm of 𝐻=𝐿2(𝐺). We define the linear operator 𝐴=Δ with domain 𝐷(𝐴)=𝐻2𝐻(𝐺)10(𝐺). 𝐴 is positive and selfadjoint. By spectral theory, we can define the powers 𝐴𝑠 and spaces 𝐻𝑠=𝐷(𝐴𝑠/2) with norms |𝑢|𝑠=|𝐴𝑠/2𝑢| for real 𝑠. Note that 𝐻0=𝐿2(𝐺). It is well known that 𝐻𝑠 is a subspace of 𝐻𝑠(𝐺) and ||𝑠 is on 𝐻𝑠(𝐺) a norm equivalent to the usual one. Moreover, we have the following Poincare inequality and interpolation inequality: |𝑢|𝑠1𝜆(𝑠2𝑠11)/2|𝑢|𝑠2,𝑠1,𝑠2𝑅,𝑠1<𝑠2,𝑢𝐻𝑠2,(2.1)|𝑢|𝜎𝑠1+(1𝜎)𝑠2|𝑢|𝜎𝑠1|𝑢|𝑠1𝜎2[],,𝜎0,1(2.2) where 𝜆1 is the first eigenvalue of 𝐴.

We can define 𝐴1𝐻𝐷(𝐴) to be the Green’s operator for 𝐴. Thus, 𝑣=𝐴1𝑤𝐴𝑣=𝑤.(2.3) By Rellich’s Theorem, we know that 𝐴1 is compact, and 𝐴𝐷(𝐴)𝐻 is a linear and bounded operator. Finally, we introduce the invertible operator 𝐵𝛼𝐻𝑠𝐻𝑠, 𝑠 defined by 𝐵𝛼=𝛼𝐼+(1𝛼)𝐴1.(2.4) For each 𝛼(0,1] and 𝛽0, we know that 𝐵𝛽𝛼𝐻𝑠𝐻𝑠 is bounded and has a bounded inverse (see [10, 11]). We also define the operator 𝐴𝛼=𝐵𝛼1𝐴 with domain𝐷𝐴𝛼=𝐷𝐴𝐷(𝐴)if𝛼>0,0=𝐻4.(2.5) By definition, it is clear that 𝐷(𝐴𝛼𝑠/2)=𝐻𝑠 in case 𝛼>0.

Lemma 2.1. For 𝛼>0, there exist 𝑀1,𝑀2, and 𝑀3 such that 𝛼1/2|𝑣||𝑣|𝐵𝛼𝑀11/2𝛼|𝑣|,𝑣𝐻,(2.6)1/2|𝑣|1|𝑣|1,𝐵𝛼𝑀21/2|𝑣|1,𝑣𝐻1𝜆,(2.7)1𝛼𝜆1+1𝛼1/2|𝑣||𝑣|𝐵𝛼1𝑀31/2|𝑣|,𝑣𝐻,(2.8) where |𝑣|𝐵𝛼=𝑣,𝐵𝛼𝑣1/2,|𝑣|1,𝐵𝛼𝐴=1/2𝑣,𝐵𝛼𝐴1/2𝑣1/2,|𝑣|𝐵𝛼1=𝑣,𝐵𝛼1𝑣1/2.(2.9)

Proof. Here, we only verify (2.8) is valid; the proofs of (2.6) and (2.7) can be found in [11]. Since 𝐵𝛼1/2 is bounded, there exists 𝑀30, such that |𝐵𝛼1/2|2𝑀3. Then, for any 𝑣𝐻, we have 𝑣,𝐵𝛼1𝑣=𝐵𝛼1/2𝑣,𝐵𝛼1/2𝑣=||𝐵𝛼1/2𝑢||2𝑀3|𝑣|2,(2.10) which completes the right part of (2.8).
Now, we proof the left part of (2.8) let 0<𝜆1𝜆2𝜆𝑘(2.11) denote the eigenvalues of 𝐴, repeated with the respective multiplicity, and the corresponding unit eigenvector is denoted by {𝑤𝑘}𝑘=1, which forms an orthonormal basis for 𝐻. We have 𝑤𝑘,𝐵𝛼1𝑤𝑘=𝜆𝑘𝛼𝜆𝑘𝜆+1𝛼1𝛼𝜆1+1𝛼,𝑘+.(2.12) Since 𝑣𝐻, there exist {𝑏𝑘}𝑘=1, such that 𝑣=+𝑘=1𝑏𝑘𝑤𝑘. Consequently, 𝑣,𝐵𝛼1𝑣=+𝑘=1𝑏𝑘𝑤𝑘,𝐵𝛼1+𝑘=1𝑏𝑘𝑤𝑘=+𝑘=1𝑏𝑘𝑤𝑘,𝐵𝛼1𝑏𝑘𝑤𝑘=+𝑘=1𝜆𝑘𝛼𝜆𝑘𝑏+1𝛼2𝑘𝜆1𝛼𝜆1+1𝛼+𝑘=1𝑏2𝑘=𝜆1𝛼𝜆1+1𝛼|𝑣|2,(2.13) which finishes the proof.

2.2. Assumptions on the Noise

The stochastic process 𝑊(𝑡), defined on a probability space (Ω,,𝐏), is a two-side in time Wiener process on 𝐻 which is given by the expansions 𝑊(𝑡)=𝑘=0𝛼𝑘𝛽𝑘(𝑡)𝑤𝑘,(2.14) where {𝑤𝑘}𝑘=1 is a basis of 𝐻 consisting of unit eigenvectors of 𝐴, {𝛼𝑘}𝑘=1 is a bounded sequence of nonnegative numbers, and 𝛽𝑘1(𝑡)=𝛼𝑘𝑊(𝑡),𝑤𝑘,𝑘(2.15) is a sequence of mutually independent real valued standard Brownian motions in a fixed probability space (Ω,,𝐏) adapted to a filtration {𝑡}𝑡0.

For convenience, we will define the covariance operator 𝑄 on 𝐻 as follows: 𝑄𝑤𝑘=𝛼𝑘𝑤𝑘,𝑘.(2.16) The process 𝑊(𝑡) will be called as the 𝑄-Wiener process. We need to impose on 𝑄 one of the following assumptions: (Q1)Tr[𝐵𝛼1𝛿𝐴2+𝛿𝑄]<(forsome0<𝛿1), (Q1*)Tr[𝐵𝛼2𝐴1𝑄]<,and Tr[𝐵𝛼2𝐴2𝑄]2𝐷,(Q2)Tr[𝐵𝛼1𝛿𝐴1+𝛿𝑄]<(for some0<𝛿1),Tr[𝐵𝛼2𝑄]<, and Tr[𝐵𝛼2𝐴2𝑄]2𝐷,(Q2*)Tr[𝐵𝛼1𝛿𝐴1+𝛿𝑄]<,Tr[𝐵𝛼2𝐴𝜎𝑄]< (for some 0<𝛿1 and 𝜎>0), and Tr[𝐵𝛼2𝐴2𝑄]2𝐷,

where 𝐷 is given in Section 4. It is obvious that 𝐐𝟐(𝐐𝟐),𝐐𝟏(𝐐𝟏).(2.17)

2.3. Main Results

We will assume throughout the paper that the space dimension 𝑛 and the integer 𝑝 in (1.3) satisfy the following growth condition: 𝑝=anypositiveinteger,if𝑛=1or2,2,if𝑛=3.(2.18)

Under the above assumptions on the noise, we can now put the original problem (1.1)-(1.2) in an abstract form 𝐴𝑑𝑢+𝛼𝑢+𝐵𝛼1𝑓(𝑢)𝑑𝑡=𝐵𝛼1𝐴1𝑑𝑊,(2.19) with which we will also associate the following initial condition: 𝑢𝑡0=𝑢0.(2.20) Note that since 𝐵𝛼1 is bounded from 𝐻𝑠 into itself for each 𝛼>0, (2.19) is qualitatively of second order in space for 𝛼>0 although it also has a nonlocal character. In contrast, for 𝛼=0 the equation is of fourth-order in space and local in character. Thus, 𝛼=0 is a singular limit for the equation.

Definition 2.2. Let 𝐼=[𝑡0,𝑡0+𝜏) be an interval in . We say that a stochastic process 𝑢(𝑡,𝜔;𝑡0,𝑢0) is a mild solution of the system (2.19)-(2.20) in 𝐻𝑠, if 𝑢,𝜔;𝑡0,𝑢0𝐶𝐼;𝐻𝑠,𝐏-a.s.𝜔Ω,(2.21) moreover, it satisfies in 𝐻𝑠 the following integral equation: 𝑢𝑡,𝜔;𝑡0,𝑢0=𝑒𝐴𝛼(𝑡𝑡0)𝑣0𝑡𝑡0𝑒𝐴𝛼(𝑡𝑠)𝐵𝛼1𝑓(𝑢)𝛽𝑊𝐴(𝑠)𝑑𝑠+𝑊𝐴(𝑡),𝐏-a.s.𝜔Ω,(2.22) where 𝑊𝐴(𝑡) is called stochastic convolution (see Section 3 for details), 𝛽 is a positive constant chosen in Section 3 and 𝑣0=𝑢0𝑊𝐴(𝑡0).

The main results of the paper are contained in the following two theorems.

Theorem 2.3. (𝑖) Let 𝛼=0, and, the hypothesis (𝐐𝟐) be satisfied. Then for every 𝑢0𝐻2, there is a unique maximally defined mild solution 𝑢(𝑡,𝜔;𝑡0,𝑢0) of (2.19)-(2.20) in 𝐻2 for all 𝑡[𝑡0,).
(𝑖𝑖) Let 𝛼(0,1], and, the hypothesis (𝐐𝟏) be satisfied. Then for every 𝑢0𝐻1, there is a unique maximally defined mild solution 𝑢(𝑡,𝜔;𝑡0,𝑢0) of (2.19)-(2.20) in 𝐻1 for all 𝑡[𝑡0,).

Theorem 2.4. (i) Let 𝛼=0, and, the hypothesis (𝐐𝟐) be satisfied. Then the stochastic flow associated with (2.19)-(2.20) has a compact stochastic attractor 𝒜0(𝜔)𝐻2 at time 0, which pullback attracts every bounded deterministic set 𝐵𝐻2.
(ii) Let 𝛼(0,1], and, the hypothesis (𝐐𝟏) be satisfied. Then the stochastic flow associated with (2.19)-(2.20) has a compact stochastic attractor 𝒜𝛼(𝜔)𝐻1 at time 0, which pullback attracts every bounded deterministic set 𝐵𝐻1.

3. Stochastic Convolution

Let 𝑊𝐴(𝑡) be the unique solution of linear equation 𝐴𝑑𝑢+𝛼+𝛽𝑢𝑑𝑡=𝐵𝛼1𝐴1𝑑𝑊,(3.1) where 𝛽 is a positive constant to be further determined. Then, 𝑊𝐴(𝑡) is an ergodic and stationary process [9, 12] called the stochastic convolution. Moreover, 𝑊𝐴(𝑡)=𝑡𝑒(𝑡𝑠)(𝐴𝛼+𝛽)𝐵𝛼1𝐴1𝑑𝑊(𝑠).(3.2)

Some regularity properties satisfied by 𝑊𝐴(𝑡) are given below.

Lemma 3.1. Assume that (𝐐𝟏) holds. Then, 𝑊𝐴(𝑡) has a version which is 𝛾-Hölder continuous with respect to (𝑡,𝑥)×𝐺 for any 𝛾[0,𝛿/2).

Proof. We only consider the case 𝑛=3. For the sake of simplicity, we also assume that 𝐺=3𝑖=1(0,𝜋). The eigenvectors of 𝐴 can be given explicitly as follows: 𝑤𝑘2(𝑥)=𝜋3/2cos𝑘1𝑥1cos𝑘2𝑥2cos𝑘3𝑥3𝑥,𝑥=1,𝑥2,𝑥33,(3.3) with corresponding eigenvalues 𝜇𝑘=𝑘21+𝑘22+𝑘23=||𝑘||2,𝑘+3,(3.4) where 𝑘=(𝑘1,𝑘2,𝑘3) varies in (+)3. Using (2.14), we find that 𝑊𝐴(𝑡,𝑥)=𝑘(+)3𝛼𝑘𝑡𝑒(𝑡𝑠)(𝜂𝑘+𝛽)1𝛼𝜇𝑘+1𝛼𝑑𝛽𝑘(𝑤𝑠)𝑘(𝑥),(3.5) where 𝜂𝑘=𝜇2𝑘/(𝛼𝜇𝑘+1𝛼), and hence, 𝑊𝐴(𝑡,𝑥)𝑊𝐴=(𝑡,𝑦)𝑘(+)3𝛼𝑘𝑡𝑒(𝑡𝑠)(𝜂𝑘+𝛽)(1/(𝛼𝜇𝑘+1𝛼))𝑑𝛽𝑘(𝑠)𝑤𝑘(𝑥)𝑤𝑘(,𝐄||𝑦)𝑊𝐴(𝑡,𝑥)𝑊𝐴||(𝑡,𝑦)2𝑘+3𝛼𝑘𝛼𝜇𝑘+1𝛼2𝑡𝑒2(𝑡𝑠)(𝜂𝑘+𝛽)||𝑑𝑠𝑤𝑘(𝑥)𝑤𝑘||(𝑦)2.(3.6)
For any 𝛾[0,1], one trivially verifies that there is a constant 𝑐𝛾>0 independent of 𝑘 such that for any 𝑘(+)3 and 𝑥,𝑦𝐺||𝑤𝑘(𝑥)𝑤𝑘||(𝑦)𝑐𝛾𝜇𝑘(1+𝛾)/2||||𝑥𝑦𝛾.(3.7) Thus, we have 𝐄||𝑊𝐴(𝑡,𝑥)𝑊𝐴||(𝑡,𝑦)2𝑐2𝛾2||||𝑥𝑦2𝛾𝑘+3𝛼𝑘𝛼𝜇𝑘+1𝛼2𝜂𝑘1𝜇𝑘1+𝛾=𝑐2𝛾2||||𝑥𝑦2𝛾𝑘+3𝛼𝑘𝛼𝜇𝑘+1𝛼2𝛼𝜇𝑘+1𝛼𝜇2𝑘𝜇𝑘1+𝛾=𝑐2𝛾2||||𝑥𝑦2𝛾𝑘+3𝛼𝑘𝜇𝑘𝛼𝜇𝑘𝜇+1𝛼𝑘2+𝛾.(3.8)
Now, let 𝑡,𝑠. We may assume that 𝑡𝑠. Then, 𝐄||𝑊𝐴(𝑡,𝑥)𝑊𝐴||(𝑠,𝑥)2=𝑘+3𝛼𝑘𝛼𝜇𝑘+1𝛼2×𝑡𝑠𝑒2(𝜂𝑘+𝛽)(𝑡𝜎)𝑑𝜎+𝑠𝑒(𝜂𝑘+𝛽)(𝑡𝜎)𝑒(𝜂𝑘+𝛽)(𝑠𝜎)2||𝑑𝜎𝑤𝑘||(𝑥)2=𝑘+3𝛼𝑘𝛼𝜇𝑘+1𝛼212𝜂𝑘+𝛽1𝑒2(𝜂𝑘+𝛽)(𝑡𝑠)||𝑤𝑘||(𝑥)2.(3.9) Let 0𝛾1/2, and let 𝑐𝛾=sup𝑟1,𝑟20||𝑒𝑟1𝑒𝑟2||||𝑟1𝑟2||2𝛾.(3.10) Since the function g(𝑟)=𝑒𝑟 is a Lipschitzoneon [0,), we always have 𝑐𝛾<. Observe that 𝐄||𝑊𝐴(𝑡,𝑥)𝑊𝐴||(𝑠,𝑥)24𝛾𝜋3𝑐𝛾|𝑡𝑠|2𝛾𝑘(+)3𝛼𝑘𝛼𝜇𝑘+1𝛼2𝜂𝑘+𝛽2𝛾1𝜇𝑘.4𝛾𝜋3𝑐𝛾|𝑡𝑠|2𝛾𝑘(+)3𝛼𝑘𝛼𝜇𝑘+1𝛼2𝜂𝑘2𝛾1𝜇𝑘=4𝛾𝜋3𝑐𝛾|𝑡𝑠|2𝛾𝑘(+)3𝛼𝑘𝜇𝑘𝛼𝜇𝑘+1𝛼2𝛾+1𝜇𝑘2+2𝛾.(3.11) By (𝐐𝟏), we know that Tr[𝐵𝛼1𝛿𝐴2+𝛿𝑄]< for some 0<𝛿1. Therefore, by (3.8) and (3.11), one deduces that there exists a constant 𝑐𝛾>0 such that 𝐄||𝑊𝐴(𝑡,𝑥)𝑊𝐴||(𝑠,𝑦)2𝑐𝛾||||𝑥𝑦2+|𝑡𝑠|2𝛾,(𝑡,𝑥),(𝑠,𝑦)×𝐺.(3.12) As 𝑊𝐴(𝑡,𝑥)𝑊𝐴(𝑠,𝑦) is a Gaussian process, we find that for each 𝑚+, there is a constant 𝑐𝑚𝛾>0 such that 𝐄||𝑊𝐴(𝑡,𝑥)𝑊𝐴||(𝑠,𝑦)2𝑚𝑐𝑚𝛾||||𝑥𝑦2+|𝑡𝑠|2𝑚𝛾.(3.13) Now, thanks to the well-known Kolmogorov test, one concludes that 𝑊𝐴(𝑡,𝑥) is (𝛾2/𝑚)-Hölder continuous in (𝑡,𝑥). Because 𝛾[0,1/2] and 𝑚+ are arbitrary, we see that the conclusion of the lemma holds true. The proof is complete.

Lemma 3.2. Assume (𝐐𝟐) holds. Then, for any 𝑀>0, there exists a 𝛽0 such that for all 𝛽𝛽0, 𝐄||𝑊𝐴||(𝑡)22𝑀.(3.14)

Proof. 𝐄||Δ𝑊𝐴(||𝑡)2=𝐄𝑘(+)3𝛼𝑘𝑡𝑒(𝜂𝑘+𝛽)(𝑡𝑠)1𝛼𝜇𝑘+1𝛼𝑑𝛽𝑘(𝑠)Δ𝑤𝑘(𝑥)2=𝑘(+)3𝛼𝑘𝛼𝜇𝑘+1𝛼2𝑡𝑒2(𝜂𝑘+𝛽)(𝑡𝑠)||𝑑𝑠Δ𝑤𝑘||(𝑥)2𝑘+3𝛼𝑘𝛼𝜇𝑘+1𝛼212𝜂𝑘||+𝛽Δ𝑤𝑘||(𝑥)212𝜂1+𝛽𝑘(+)3𝛼𝑘𝜇𝑘𝛼𝜇𝑘+1𝛼2.(3.15) Since Tr[𝐵𝛼2𝑄]<, one can now easily choose a 𝛽 large enough so that 𝐄(|Δ𝑊𝐴(𝑡)|2)𝑀, and the proof is complete.

Similarly, we can verify the following basic fact.

Lemma 3.3. Assume (𝐐𝟐) holds. Then, Δ𝑊𝐴 has a version which is 𝛾-Hölder continuous with respect to (𝑡,𝑥)×𝐺 for any 𝛾[0,𝛿/2).

Lemma 3.4. Assume that (𝐐𝟐) holds. Then, for any 𝑀>0, there exists 𝛽0 such that for all 𝛽𝛽0, 𝐄||𝑊𝐴||(𝑡)22+𝜎𝑀.(3.16)

4. Stochastic dissipativeness in 𝐻1

It is well known that in the deterministic case without forcing terms, 1𝐽(𝑢)=2||||𝑢2+𝐺𝐹(𝑢)𝑑𝑥(4.1) is a Lyapunov functional of the system (i.e. (𝑑/𝑑𝑡)𝐽(𝑢)0), where 𝐹(𝑢) is the primitive function of 𝑓(𝑢) which vanishes at zero. In this section, we will prove a similar property for the stochastic equation by adapting some argument in [9].

Assume that 𝑢 satisfies (2.19)-(2.20). As usual, we may assume in advance that 𝑢 is sufficiently regular so that all the computations can be performed rigorously. Applying the Itô formula to 𝐽(𝑢), we obtain 𝐽𝑑𝐽(𝑢)=𝑢+1(𝑢),𝑑𝑢2𝐽Tr𝑢𝑢(𝑢)𝐵𝛼2𝐴2𝑄=𝐽𝑑𝑡𝑢(𝑢),𝐵𝛼1𝐴1𝐽𝑑𝑊𝑢(𝑢),𝐵𝛼1𝐴𝑢+𝐵𝛼11𝑓(𝑢)𝑑𝑡+2𝐽Tr𝑢𝑢(𝑢)𝐵𝛼2𝐴2𝑄𝑑𝑡,(4.2) where 𝐽𝑢,𝐽𝑢𝑢 denote, respectively, the first and second derivative of 𝐽. Since 𝐽𝑢(𝑢)=𝐴𝑢+𝑓(𝑢),(4.3) there exists 𝐶1>0 such that for 𝛼=0, 𝐽𝑢(𝑢),𝐵𝛼1𝐴𝑢+𝐵𝛼1=||||𝑓(𝑢)𝐴𝑢+𝑓(𝑢)21𝜆21||||𝐴𝑢+𝑓(𝑢)21=𝜆21𝐴𝑢+𝑓(𝑢),𝑢+𝐴1𝑓(𝑢)=𝜆21|𝑢|21+||||𝑓(𝑢)21+2(𝑓(𝑢),𝑢)𝑑𝜆21|𝑢|21+𝐺𝐹(𝑢)𝑑𝑥𝐶1=𝑑𝜆21𝐽(𝑢)𝐶1,(4.4) where 𝑑=min{1,4𝑝𝑎2𝑝1}. And for 0<𝛼1, 𝐽𝑢(𝑢),𝐵𝛼1𝐴𝑢+𝐵𝛼1=𝑓(𝑢)𝐴𝑢+𝑓(𝑢),𝐵𝛼1𝐴𝑢+𝐵𝛼1=||||𝑓(𝑢)𝐴𝑢+𝑓(𝑢)2𝐵𝛼1𝜆21𝛼𝜆1||||+1𝛼𝐴𝑢+𝑓(𝑢)21𝑑𝜆21𝛼𝜆1+1𝛼𝐽(𝑢)𝐶1,(4.5) where we have used (2.8). Simple computations show that 𝐽𝑢𝑢(𝑢)=𝐴+𝑓(𝑢),(4.6) and hence, 𝐽Tr𝑢𝑢(𝑢)𝐵𝛼2𝐴2𝑄=Tr𝐴𝐵𝛼2𝐴2𝑄+𝑖=1𝐷𝑖𝐺𝑓(𝑢)𝑤2𝑖𝐵𝑑𝑥=Tr𝛼2𝐴1𝑄+𝑖=1𝐷𝑖𝐺𝑓(𝑢)𝑤2𝑖,𝑑𝑥(4.7) where {𝑤𝑖}𝑖=1 is the orthonormal basis of 𝐻 as in (2.14), and 𝐷𝑖=𝛼𝑖/(𝛼𝜆𝑖+1𝛼)2.

We infer from (3.3) that ||𝑤𝑖||𝐿𝐶2,(4.8) where 𝐶2>0 depends only on 𝐺. Therefore, ||||𝐺𝑓(𝑢)𝑤2𝑖||||𝑑𝑥𝐶22𝐺||𝑓||(𝑢)𝑑𝑥.(4.9) Set 𝐶3 such that ||𝑓||(𝑠)2(2𝑝1)𝑎2𝑝1𝑠2𝑝2+𝐶3,𝑠,(4.10) then||||𝐺𝑓(𝑢)𝑤2𝑖||||𝑑𝑥𝐶222(2𝑝1)𝑎2𝑝1𝐺𝑢2𝑝2𝑑𝑥+𝐶3||𝐺||1𝑎4𝑝2𝑝1𝐺𝑢2𝑝𝑑𝑥+𝐶4,(4.11) where 𝐶4 depends on 𝑓, 𝑝, and 𝐺. Let 𝐶5 satisfy 1𝐹(𝑠)𝑎4𝑝2𝑝1𝑠2𝑝𝐶5||𝐺||,𝑠,(4.12) then ||||𝐺𝑓(𝑢)𝑤2𝑖||||𝑑𝑥𝐽(𝑢)+𝐶4+𝐶5.(4.13) Finally, 𝐽Tr𝑢𝑢(𝑢)𝐵𝛼2𝐴2𝑄𝐵Tr𝛼2𝐴1𝑄𝐵+Tr𝛼2𝐴2𝑄𝐽(𝑢)+𝐶4+𝐶5.(4.14) Since 𝐄𝐽𝑢(𝑢),𝐵𝛼1𝐴1𝑑𝑊=0,(4.15) we have from (4.2) that 𝑑𝐽𝑑𝑡𝐄(𝐽(𝑢))=𝐄𝑢(𝑢),𝐵𝛼1𝐴(𝑢)𝐵𝛼1+1(𝑢)2𝐄𝐽Tr𝑢𝑢(𝑢)𝐵𝛼2𝐴2𝑄.(4.16) Further, by (4.4), (4.5) and (4.14), it holds that𝑑1𝑑𝑡𝐄(𝐽(𝑢))𝐷2𝐵Tr𝛼2𝐴2𝑄𝐵𝐄(𝐽(𝑢))+Tr𝛼2𝐴1𝑄𝐵+Tr𝛼2𝐴2𝑄𝐶4+𝐶5+𝐶1,(4.17) where 𝐷=min{𝑑𝜆21,𝑑𝜆21/(𝛼𝜆1+1𝛼)}. This is precisely what we promised.

Now, by directly applying the classical Gronwall Lemma, we have the following lemma.

Lemma 4.1. Let 𝑊 be a H-valued Q-Wiener process with 𝐵Tr𝛼2𝐴1𝑄𝐵<+,Tr𝛼2𝐴2𝑄2𝐷,(4.18) and let 𝑢(𝑡) be the mild solution to (2.19)-(2.20). Then, 𝐄𝐽𝑢(𝐽(𝑢(𝑡)))𝐄0+𝐶𝑄𝑡,𝑡0,,(4.19) where 𝐶𝑄=𝐵Tr𝛼2𝐴1𝑄𝐵+Tr𝛼2𝐴2𝑄𝐶4+𝐶5+𝐶1𝐵𝐷(1/2)Tr𝛼2𝐴2𝑄.(4.20)

As a consequence, we immediately obtain the following basic result.

Corollary 4.2. Let 𝑊 be a H-valued Q-Wiener process with 𝐵Tr𝛼2𝐴1𝑄𝐵<+,Tr𝛼2𝐴2𝑄2𝐷.(4.21)
Then, there exists a continuous nonnegative function Ψ(𝑟) such that for any solution 𝑢(𝑡) of (2.19)-(2.20), one has 𝐄||||𝑢(𝑡)21𝐄||𝑢Ψ0||21𝑡,𝑡0,.(4.22)

5. The Existence and Unique of Global Mild Solutions

In this section, we study the existence and unique of global mild solutions of the problem (2.19)-(2.20). The basic idea is to transform the original problem into a nonautonomous one by using the simple variable change below: 𝑣(𝑡)=𝑢(𝑡)𝑊𝐴(𝑡).(5.1)

We observe that 𝑣(𝑡) satisfies the following system: 𝑑𝑣+𝐴𝑑𝑡𝛼𝛽𝑣+𝐵𝛼1𝑓𝑣+𝑊𝐴𝑣𝑡=0,0=𝑢0𝑊𝐴𝑡0.(5.2) Let 𝐺(𝑣,𝑡)=𝐵𝛼1𝑓𝑣+𝑊𝐴+𝛽𝑊𝐴,𝑣0=𝑢0𝑊𝐴𝑡0.(5.3) Then, (5.2) reads 𝑑𝑣𝑑𝑡+𝐴𝛼𝑣𝑡𝑣=𝐺(𝑣,𝑡),0=𝑣0.(5.4) To prove Theorem 2.3, it suffices to establish some corresponding existence results for the nonautonomous system (5.4).

Definition 5.1. Let 𝐼=[𝑡0,𝑡0+𝜏) be an interval in . We say that a stochastic process 𝑣(𝑡,𝜔;𝑡0,𝑣0) is a mild solution of the system (5.4) in 𝐻𝑠, if 𝑣,𝜔;𝑡0,𝑣0𝐶𝐼;𝐻𝑠,𝐏-a.s.𝜔Ω,(5.5) and satisfies in 𝐻𝑠 the following integral equation: 𝑣𝑡,𝜔;𝑡0,𝑣0=𝑒𝐴𝛼(𝑡𝑡0)𝑣0𝑡𝑡0𝑒𝐴𝛼(𝑡𝑠)𝐵𝛼1𝑓(𝑢)𝛽𝑊𝐴(𝑠)𝑑𝑠,𝐏-a.s.𝜔Ω.(5.6)

Theorem 5.2. Let 𝛼=0. Suppose that the Hypothesis (Q2) is satisfied.
Then, for every 𝑢0𝐻2, there is a unique globally defined mild solution 𝑣(𝑡,𝜔;𝑡0,𝑣0) of (5.4) in 𝐻2 with 𝑣𝑡,𝜔;𝑡0,𝑣0𝑡𝐶0,;𝐻2𝐶0,1𝑟loc𝑡0,;𝐻4𝑟𝑡𝐶0,;𝐻4,(5.7) for all 0𝑟<1.

Proof. We only consider the case where 𝑛=3. First, it is easy to verify that 𝐏-a.s. 𝐺(𝑣,𝑡)𝐶Lip;𝛾𝐻2×𝑡0,,𝐻.(5.8) Indeed, by Lemma 3.3, we see that 𝑊𝐴(𝑡)𝐻2 is 𝛾-Hölder continuous with respect to 𝑡   𝐏-a.s. Recall that 𝑓 is a polynomial of degree 2𝑝1 with 𝑝=2 (in case 𝑛=3). One deduces that there exist 𝐶1,𝐶2(𝜔)>0 such that ||𝐺𝑣1,𝑡1𝑣𝐺2,𝑡2||𝐶1||𝑣1𝑣2||2+||𝑊𝐴𝑡1𝑊𝐴𝑡2||2𝐶2||𝑣(𝜔)1𝑣2||2+||𝑡1𝑡2||𝛾,𝐏-a.s.(5.9) It then follows from [11, Lemma 47.4] that there is a unique maximally defined mild solution 𝑣 of (5.4) in 𝐻2 on [𝑡0,𝑇) satisfying 𝐏-a.s. 𝑣𝑡,𝜔;𝑡0,𝑣0=𝑒𝐴2(𝑡𝑡0)𝑣0𝑡𝑡0𝑒𝐴2(𝑡𝑠)𝐴𝑓(𝑢(𝑠))𝛽𝑊𝐴(𝑣𝑠)𝑑𝑠,𝑡,𝜔;𝑡0,𝑣0𝑡𝐶0,𝑇;𝐻2𝐶0,1𝑟loc𝑡0,𝑇;𝐻4𝑟𝑡𝐶0,𝑇;𝐻4,(5.10) for all 0𝑟<1. Furthermore, we also know that 𝑣 is a strong solution in 𝐻2. Hence, it satisfies in the strong sense that 𝑑𝑣𝑑𝑡+𝐴2𝑣+𝐴𝑓(𝑢)𝛽𝑊𝐴𝑡=0,𝑣0=𝑣0.(5.11) In what follows, we show 𝑇=, thus proving the theorem.
Simple computations yields ||||||𝑓Δ𝑓(𝑢)||(𝑢)𝐿||||+||𝑓Δ𝑢||(𝑢)𝐿||||𝑢2𝐿4.(5.12) Since 𝑓 is a polynomial of degree 3, there exist 𝜅1 and 𝜅2 such that ||𝑓||(𝑠)𝜅11+|𝑠|2,||𝑓||(𝑠)𝜅2(1+|𝑠|),𝑠.(5.13) Therefore, ||𝑓||(𝑢)𝐿||||Δ𝑢𝜅11+|𝑢|2𝐿||||Δ𝑢2𝜅11+|𝑣|2𝐿+||𝑊𝐴||2𝐿||||+||Δ𝑣Δ𝑊𝐴||.(5.14) By the Nirenberg-Gagiardo inequality, there exist 𝐶3,𝐶4,𝐶5>0 such that |𝑢|2𝐿𝐶3||||Δ𝑢2,𝑢𝐻2,|𝑢|2𝐿𝐶4||Δ2𝑢||1/3|𝑢|𝐿5/36,𝑢𝐻4,||||Δ𝑢𝐶5||Δ2𝑢||1/2||||𝑢1/2,𝑢𝐻4.(5.15) Hence, ||𝑓||(𝑢)𝐿||||Δ𝑢2𝜅11+|𝑣|2𝐿+||𝑊𝐴||2𝐿||||+||Δ𝑣Δ𝑊𝐴||2𝜅11+𝐶4||Δ2𝑣||1/3|𝑣|𝐿5/36+𝐶3||Δ𝑊𝐴||2𝐶5||||𝑣1/2||Δ2𝑣||1/2+||Δ𝑊𝐴||.(5.16) By (Q2) and Lemma 3.2, we know that for 𝐏-a.s. 𝜔Ω, there exists an 𝑅1(𝜔)>0 such that |Δ𝑊𝐴(𝑡)|𝑅1(𝜔) (for all 𝑡). On the other hand, by Lemma 3.2 and Corollary 4.2, we find that 𝐏-a.s. 𝑣 is bounded in 𝐻1. Thus, for 𝐏-a.s. 𝜔Ω, there exist 𝐶6(𝜔),𝐶7(𝜔)>0 such that |𝑣|𝐿5/36𝐶6||||(𝜔),𝑣1/2𝐶7(𝜔),(5.17) where the continuous imbedding 𝐻1𝐿6 is used. Consequently, we have ||𝑓||(𝑢)𝐿||||Δ𝑢𝐶8||Δ(𝜔)1+2𝑣||1/3+𝑅1||Δ(𝜔)2𝑣||1/2+𝑅1(𝜔),𝐏-a.s.𝜔Ω.(5.18)
Similarly for 𝐏-a.s. 𝜔Ω, one easily deduces that there exists 𝐶9(𝜔)>0 such that ||𝑓||(𝑢)𝐿||||Δ𝑢𝐶9||Δ(𝜔)1+2𝑣||1/6+𝑅1||Δ(𝜔)2𝑣||1/4+𝑅1(𝜔).(5.19) It then follows from (5.12) that for 𝐏-a.s. 𝜔Ω, ||||Δ𝑓(𝑢)𝐶8||Δ(𝜔)1+2𝑣||1/3+𝑅1||Δ(𝜔)2𝑣||1/2+𝑅1(𝜔)+𝐶9(𝜔)1+𝐿3||Δ2𝑣||1/6+𝑅1||Δ(𝜔)2𝑣||1/4+𝑅1(𝜔)𝐶10||Δ(𝜔)1+2𝑣||5/6.(5.20)
Now, taking the 𝐿2 inner-product of equation (5.11) with Δ2𝑣, one obtains 12𝑑||||𝑑𝑡Δ𝑣2+||Δ2𝑣||2||||𝐺Δ𝑓(𝑢)Δ2||||||||𝑣𝑑𝑥+𝛽𝐺𝑊𝐴Δ2||||1𝑣𝑑𝑥4||Δ2𝑣||2+||||Δ𝑓(𝑢)2+14||Δ2𝑣||2+𝛽2||𝑊𝐴||212||Δ2𝑣||2+||||Δ𝑓(𝑢)2+𝛽2𝜆12||Δ𝑊𝐴||2.(5.21) By (5.20), we deduce that 𝐏-a.s. 𝑑||||𝑑𝑡Δ𝑣2+||Δ2𝑣||2𝐶11||Δ(𝜔)1+2𝑣||5/3.(5.22) Furthermore, by Young's inequality and |Δ2𝑣|2𝜆21|Δ𝑣|2, we know that there exists 𝐶12(𝜔)>0 such that 𝐏-a.s. 𝑑||||𝑑𝑡Δ𝑣2𝜆212||||Δ𝑣2+𝐶12(𝜔).(5.23) Applying the gronwall lemma on (5.23), one gets ||||Δ𝑣22𝐶12(𝜔)𝜆21,𝐏-a.s.𝜔Ω.(5.24) This implies that the weak solution solution 𝑣 does not blow up in finite time in the space 𝐻2. Hence, 𝑇(𝑣0)=, for all 𝑢0𝐻2.

Theorem 5.3. Let 𝛼(0,1], and let Hypothesis (Q1) be satisfied. Then, for every 𝑢0𝐻1, there is a unique maximally defined mild solution 𝑣(𝑡,𝜔;𝑡0,𝑣0) of (5.4) in 𝐻1 for all 𝑡[𝑡0,) with 𝑣𝑡,𝜔;𝑡0,𝑣0𝑡𝐶0,;𝐻1𝐶0,1𝑟loc𝑡0,;𝐻2𝑟𝑡𝐶0,;𝐻2,(5.25) for 0𝑟<1.

Proof. As noted above, 𝐴𝛼 is a positive selfadjoint linear operator on 𝐻 with compact resolvent. The negative operator 𝐴𝛼 generate an analytic semigroup 𝑒𝐴𝛼𝑡. It is easy to verify by Lemma 3.1 that 𝐏-a.s. 𝐺(𝑣,𝑡)𝐶Lip;𝛾𝐻1×𝑡0,,𝐻.(5.26) It then follows from [11, Lemma 47.4] that there is a unique maximally defined mild solution 𝑣 of (5.4) in 𝐻1 on [𝑡0,𝑇) with 𝑣𝑡,𝜔;𝑡0,𝑣0𝑡𝐶0,𝑇;𝐻1𝐶0,1𝑟loc𝑡0,𝑇;𝐻2𝑟𝑡𝐶0,𝑇;𝐻2,(5.27) where 𝑡0<𝑇=𝑇(𝑣0), and 0𝑟<1. Furthermore, 𝑣 is a strong solution in 𝐻1 and hence solves (5.4) in the strong sense. To complete the proof of the theorem, there remains to check that 𝑇(𝑣0)=.
Equation (5.4) is equivalent to 𝐵𝛼𝑑𝑣𝑑𝑡+𝐴𝑣=𝑓𝑣+𝑊𝐴+𝛽𝐵𝛼𝑊𝐴𝑡,𝑣0=𝑣0.(5.28) Multiplying (5.28) by 𝐴𝑣, one gets 12𝑑𝑑𝑡|𝑣|21,𝐵𝛼+|𝑣|22+𝑓𝑣+𝑊𝐴𝐵,𝐴𝑣+𝛽𝛼𝑊𝐴,𝐴𝑣=0.(5.29) We observe that 𝑓𝑣+𝑊𝐴=,𝐴𝑣𝐺𝑓𝑣+𝑊𝐴=𝑣𝑑𝑥𝐺𝑓𝑣+𝑊𝐴||||𝑣2𝑑𝑥+𝐺𝑓𝑣+𝑊𝐴𝑊𝐴𝑣𝑑𝑥.(5.30) We take 𝐶1 and 𝐶2 such that 𝑓(𝑥)2𝑝12𝑎2𝑝1𝑥2𝑝2𝐶1,||𝑓(||𝑥)2(2𝑝1)𝑎2𝑝1𝑥2𝑝2+𝐶2,(5.31) for all 𝑥. Then, 𝑓𝑣+𝑊𝐴,𝐴𝑣2𝑝12𝑎2𝑝1𝐺||𝑣+𝑊𝐴||2𝑝2||||𝑣2𝑑𝑥𝐶1𝐺||||𝑣2𝑑𝑥2(2𝑝1)𝑎2𝑝1𝐺||𝑣+𝑊𝐴||2𝑝2||||||𝑣𝑊𝐴||𝑑𝑥𝐶2𝐺||||||𝑣𝑊𝐴||1𝑑𝑥4(2𝑝1)𝑎2𝑝1𝐺||𝑣+𝑊𝐴||2𝑝2||||𝑣2𝑑𝑥2𝐶1𝐺||||𝑣2𝑑𝑥𝐶3𝐺||𝑊𝐴||2𝑝𝑑𝑥+𝐺||𝑊𝐴||2,𝑑𝑥(5.32) where we have used Hölder's inequality, Young's inequality, and the appropriate imbeddings 𝐻1(𝐺)𝐿𝑟(𝐺) in dimension 𝑛=1𝑜𝑟2 and 3. We also know by (2.6) that there exists 𝛼𝐶𝛼𝑀1 such that 𝐵𝛼𝑊𝐴,𝐴𝑣𝐶𝛼𝐺||𝑊𝐴||2𝑑𝑥+𝐺||||𝑣2𝑑𝑥.(5.33) Combining the last two inequalities together, we deduce that there exists constants 𝐶4,𝐶5>0 such that 12𝑑|𝑑𝑡𝑣|21,𝐵𝛼+|𝑣|22+14(2𝑝1)𝑎2𝑝1𝐺||𝑣+𝑊𝐴||2𝑝2||||𝑣2𝑑𝑥2𝐶4𝐺||||𝑣2𝑑𝑥+𝐶5𝐺||𝑊𝐴||2𝑝𝑑𝑥+𝐺||𝑊𝐴||2.𝑑𝑥(5.34) In view of (2.7), there exists 𝛼𝐶𝛼𝑀1 such that 12𝐶𝛼𝑑|𝑑𝑡𝑣|21+|𝑣|22+14(2𝑝1)𝑎2𝑝1𝐺||𝑣+𝑊𝐴||2𝑝2||||𝑣2𝑑𝑥2𝐶4|𝑣|21+𝐶5𝐺||𝑊𝐴||2𝑝𝑑𝑥+𝐺||𝑊𝐴||2.𝑑𝑥(5.35) Using the gronwall lemma on (5.35), the following inequality holds: |𝑣|212𝑒4𝐶4/𝐶𝛼𝐶4||𝑣0||21+2𝑒4𝐶4/𝐶𝛼𝑡𝑡0𝐶5𝐺||𝑊𝐴(||𝑠)2𝑝𝑑𝑥+𝐺||𝑊𝐴(||𝑠)2𝑑𝑥𝑑𝑠.(5.36) Lemma 3.1 guarantees that 𝐏-a.s. 𝑡𝑡0𝐺||𝑊𝐴(||𝑠)2𝑝𝑑𝑥𝑑𝑠<+,𝑡𝑡0𝐺||𝑊𝐴(||𝑠)2𝑑𝑥𝑑𝑠<+.(5.37) This and (5.36) implies that the mild solution 𝑣 does not blow up in finite time in the space 𝐻1. It follows that 𝑇(𝑣0)=. The proof is complete.

Remark 5.4. The conclusions in Theorem 2.3 are readily implied in the above two theorems.

6. Attractors for Stochastic Viscous Cahn-Hilliard Equation

For convenience of the reader, some basic knowledge of RDS are summarized in the Appendix at the end of this paper.

6.1. Stochastic Flows

Thanks to Theorem 2.3, the mapping 𝑢0𝑢(𝑡,𝜔;𝑡0,𝑢0) defines a stochastic flow 𝑆𝛼(𝑡,𝑠;𝜔),𝑆𝛼(𝑡,𝑠;𝜔)𝑢0=𝑢𝑡,𝜔;𝑠,𝑢0[],𝛼0,1.(6.1) Notice that 𝐏-a.s. (i)𝑆𝛼(𝑡,𝑠;𝜔)=𝑆𝛼(𝑡,𝑟;𝜔)𝑆𝛼(𝑟,𝑠;𝜔),forall𝑠𝑟𝑡,(ii)𝑆0(𝑡,𝑠;𝜔) is continuous in 𝐻2, and 𝑆𝛼(𝑡,𝑠;𝜔) is continuous in 𝐻1 for 0<𝛼1.

6.2. Compactness Properties of Stochastic Flow 𝑆𝛼(𝑡,𝑠;𝜔)

Lemma 6.1. (i) Under Assumption (𝐐𝟐), the stochastic flow 𝑆0(𝑡,𝑠;𝜔) is uniformly compact at time 0. More precisely, for all 𝐵𝐻2 bounded and each 𝑡0<0, 𝑆0(0,𝑡0;𝜔)𝐵 is v relatively compact in 𝐻2.
(ii) Under Assumption (𝐐𝟏), the flow 𝑆𝛼(𝑡,𝑠;𝜔), 0<𝛼1, is uniformly compact at time 0. More precisely, for all 𝐵𝐻1 bounded and each 𝑡0<0, 𝑆𝛼(0,𝑡0;𝜔)𝐵 is 𝐏-a.s. relatively compact in 𝐻1.

Proof. (i) Let 𝐵𝐻2 be a given bounded deterministic set. By Lemma 3.4, we know that for 𝐏-a.s. 𝜔Ω, there exists 𝑅2(𝜔)>0, such that |𝑊𝐴(𝑡)|2+𝜎𝑅2(𝜔), 𝑡. Define 𝐵=𝐵𝐵2+𝜎(0,𝑅2(𝜔)), where 𝐵2+𝜎(0,𝑅2(𝜔)) denotes the open ball centered at 0 with radius 𝑅2(𝜔) in 𝐻2+𝜎. Then, 𝐵𝐻2 is 𝐏-a.s. bounded, and 𝑆00,𝑡0𝑒;𝜔𝐵𝐴2𝑡0𝑣00𝑡0𝑒𝐴2𝑠𝐺(𝑣(𝑠),𝑠)𝑑𝑠+𝑊𝐴(0),𝑣0𝐵𝑁1+𝑁2+𝑁3+𝑁4,(6.2)𝐏-a.s., where 𝑁1=𝑒𝐴2𝑡0𝑁𝐵,2=0𝛿𝑒𝐴2𝑠𝐺(𝑣(𝑠),𝑠)𝑑𝑠,𝑣0𝐵,𝑁3=𝑒𝐴2𝛿𝑡𝛿0𝑒𝐴2(𝑠+𝛿)𝐺(𝑣(𝑠),𝑠)𝑑𝑠,𝑣0𝐵,𝑁4=𝐵2+𝜎0,𝑅2,(𝜔)(6.3) and 𝛿 is an arbitrary constant satisfying 0<𝛿<𝑡0.

Since for 𝑡>0 fixed the operator 𝑒𝐴2𝑡 is compact, we see that 𝑁1, 𝑁3, and 𝑁4 are relatively compact sets in 𝐻2. Now, we show that 𝐏-a.s. 𝑆0(0,𝑡0;𝜔)𝐵 is relatively compact. To this end, we first give an estimate on the Kuratowski measure of 𝑁2𝐻2.

For 𝑣0𝐵, one has ||||0𝛿𝑒𝐴2(𝑠𝑡0)||||𝐺(𝑣(𝑠),𝑠)𝑑𝑠2=||||0𝛿𝐴𝑒𝐴2(𝑠𝑡0)||||.𝐺(𝑣(𝑠),𝑠)𝑑𝑠(6.4) Since 𝐴2 is a positive sectorial operator on 𝐻, there exists a constant 𝑀𝐴>0 such that |||𝐴𝑒𝐴2𝑡|||(𝐻2)𝑀𝐴𝑡1/2,𝑡0.(6.5) Recall that 𝐺(𝑣,𝑡)𝐶Lip;𝛾(𝐻2×[𝑡0,+),𝐻). So there is a 𝐾0(𝜔)>0 such that 𝐏-a.s. ||||𝐺(𝑣,𝑡)𝐾0[](𝜔),(𝑣,𝑡)𝐵×𝛿,0.(6.6) Therefore ||||0𝛿𝑒𝐴2𝑠||||𝐺(𝑣(𝑠),𝑠)𝑑𝑠2𝐾0(𝜔)𝑀𝐴0𝛿(𝑠)1/21𝑑𝑠=2𝐾0(𝜔)𝑀𝐴𝛿1/2.(6.7) It follows that 𝜅𝑁2diam𝐻2𝑁2𝐾0𝑀𝐴𝛿1/2,(6.8) where 𝜅() denotes the Kuratowski measure of noncompactness on 𝐻2. Now since 𝑁1, 𝑁3, and 𝑁4 are relatively compact sets in 𝐻2   𝐏-a.s., we have 𝜅𝑆00,𝑡0𝐵𝑁;𝜔𝜅1𝑁+𝜅2𝑁+𝜅3𝑁+𝜅4𝑁𝜅2𝐾0𝑀𝐴𝛿1/2.(6.9) Letting 𝛿0, one immediately concludes that 𝐏-a.s. 𝜅(𝑆0(0,𝑡0;𝜔)𝐵)=0, hence 𝑆0(0,𝑡0𝐵;𝜔) is relatively compact.

(ii) The proof of the compactness result for 𝑆𝛼(𝑡,𝑠;𝜔) (0<𝛼1) is fully analogous, and is thus omitted.

6.3. The Random Attractors

Now, we show that the system 𝑆𝛼(𝑡,𝑠;𝜔) possesses a random attractor 𝒜𝛼(𝜔) for every 𝛼[0,1].

Proof of Theorem 2.4. We infer from the proofs of Theorem 5.2 and Lemma 6.1 that there exists 𝑡(𝜔)<0 such that for any 𝑡0𝑡(𝜔), we can define an absorbing set for 𝑆0(𝑡,𝑡0;𝜔) at time 0 by 𝔅0=||||𝑣Δ𝑣22𝐶12(𝜔)𝜆21𝐵2+𝜎0,𝑅2(𝜔),(6.10) and for 𝑆𝛼(𝑡,𝑠;𝜔) (0<𝛼1), for any 𝑡0<0 we can define an absorbing set for 𝑆𝛼(𝑡,𝑡0;𝜔) at time 0 by 𝔅𝛼=𝐵1(0,Ψ),(6.11) where 𝐵1(0,Ψ) denotes the open ball centered at 0 with radius Ψ in 𝐻1. Now the conclusions of the theorem immediately follows from Proposition A.6


Basic knowledge of RDS

In the Appendix, we present some notations of RDS, which are also introduced in [7, 13, 14].

Let (𝑋,𝑑) be a complete metric space, and let (Ω,,𝐏) be a probability space. We consider a family of mappings {𝑆(𝑡,𝑠;𝜔)}𝑡𝑠,𝜔Ω𝑋𝑋,(A.1) satisfying 𝐏-a.s. (i)𝑆(𝑡,𝑠;𝜔)=𝑆(𝑡,𝑟;𝜔)𝑆(𝑟,𝑠;𝜔),forall𝑠𝑟𝑡,(ii)𝑆(𝑡,𝑠;𝜔) is continuous in 𝑋, for all 𝑠𝑡.

Definition A.1. We say that (𝑡,𝜔)𝑋 is an absorbing set at time 𝑡, if 𝐏-a.s. (i)(𝑡,𝜔) is bounded,(ii)for all 𝐵𝑋 there exists 𝑠𝐵 such that 𝑆(𝑡,𝑠;𝜔)𝐵(𝑡,𝜔), for all 𝑠𝑠𝐵.

Definition A.2. Given 𝑡 and 𝜔Ω, we say that {𝑆(𝑡,𝑠;𝜔)}𝑡𝑠,𝜔Ω is uniformly compact at time t if for all bounded set 𝐵𝑋, there exist 𝑠𝐵, such that 𝐏-a.s. 𝑠𝑠𝐵𝑆(𝑡,𝑠;𝜔)𝐵(A.2) is relatively compact in 𝑋.

Definition A.3. Given 𝑡 and 𝜔Ω, for any set 𝐵𝑋, we define the random omega limit set of a bounded set 𝐵𝑋 at time 𝑡 as Ω𝐵(𝑡,𝜔)=𝑇𝑡𝑠𝑇𝑆(𝑡,𝑠;𝜔)𝐵.(A.3)

Definition A.4. Let (𝑋,𝑑) be a metric space, and let {𝑆(𝑡,𝑠;𝜔)}𝑡𝑠,𝜔Ω a family of operators that maps 𝑋 into itself. We say that 𝒜(𝑡,𝜔) is a stochastic attractor if 𝐏-a.s. (i)𝒜(𝑡,𝜔) is not empty and compact,(ii)𝑆(𝜏,𝑠;𝜔)𝒜(𝑠,𝜔)=𝒜(𝜏,𝜔) for all 𝜏𝑠,(iii)for every bounded set 𝐵𝑋, lim𝑡𝑑(𝑆(𝑡,𝑠;𝜔)𝐵,𝒜(𝑡,𝜔))=0.

Remark A.5. (i) In the stochastic case, it is not possible to construct the random attractor as the Ω-limit of the absorbing set (as done in the deterministic case). This is due to the fact that the Ω-limit set is taken from and that the absorbing set is random.
(ii) Global attractor is connected.

Proposition A.6 (see [15]). If there exists a random set absorbing every bounded deterministic set 𝐵𝑋 and {𝑆(𝑡,𝑠;𝑤)}𝑡𝑠,𝜔Ω is uniformly compact at time 𝑡, then the RDS possesses a random attractor defined by 𝒜(𝑡,𝜔)=𝐵𝑋Ω𝐵(𝑡,𝜔).(A.4)

Remark A.7. In this paper, we write 𝒜(𝜔) instead of 𝒜(0,𝜔) for short.


This work was supported by NNSF of China (nos. 10732020 and 10771159).


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