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

Abstract

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: 𝑑((1−𝛼)𝑢−𝛼Δ𝑢)+Δ²𝑢−Δ𝑓(𝑢)𝑑𝑡=𝑑𝑊,(𝑥,𝑡)∈𝐺×𝑡₀,∞,(1.1) subjected to homogeneous Dirichlet boundary conditions 𝑢(𝑥,𝑡)=0,(𝑥,𝑡)∈𝜕𝐺×𝑡₀,∞,(1.2) in dimension 𝑛=1,2 or 3, where 𝐺=∏^𝑛𝑖=1(0,𝐿_𝑖) in 𝑅^𝑛, and 𝛼∈[0,1] is a parameter, 𝑓 is a polynomial of odd degree with a positive leading coefficient 𝑓(𝑥)=^2𝑝−1_𝑘=1𝑎_𝑘𝑥^𝑘,𝑎_2𝑝−1>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 𝒜_𝛼 of the system has the same finite Hausdorff dimension for different parameter values 𝛼. One can also show that 𝒜_𝛼 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)=𝐻²(𝐺)⋂𝐻¹⁰(𝐺). 𝐴 is positive and selfadjoint. By spectral theory, we can define the powers 𝐴^𝑠 and spaces 𝐻_𝑠=𝐷(𝐴^𝑠/2) with norms |𝑢|_𝑠=|𝐴^𝑠/2𝑢| for real 𝑠. Note that 𝐻₀=𝐿²(𝐺). It is well known that 𝐻_𝑠 is a subspace of 𝐻^𝑠(𝐺) and |⋅|_𝑠 is on 𝐻^𝑠(𝐺) a norm equivalent to the usual one. Moreover, we have the following Poincare inequality and interpolation inequality: |𝑢|_𝑠1≤𝜆|𝑢|^{−(𝑠2−𝑠11)/2}|𝑢|_{𝑠2𝜎𝑠1+(1−𝜎)𝑠2},∀𝑠₁,𝑠₂∈𝑅,𝑠₁<𝑠₂,∀𝑢∈𝐻_𝑠2,(2.1)≤|𝑢|^𝜎𝑠₁|𝑢|^𝑠1−𝜎₂,𝜎∈[0,1],(2.2) where 𝜆₁ is the first eigenvalue of 𝐴.

We can define 𝐴⁻¹∶𝐻→𝐷(𝐴) to be the Green’s operator for 𝐴. Thus, 𝑣=𝐴⁻¹𝑤⟺𝐴𝑣=𝑤.(2.3) By Rellich’s Theorem, we know that 𝐴⁻¹ is compact, and 𝐴∶𝐷(𝐴)→𝐻 is a linear and bounded operator. Finally, we introduce the invertible operator 𝐵_𝛼∶𝐻_𝑠→𝐻_𝑠, 𝑠∈ℝ defined by 𝐵_𝛼∶=𝛼𝐼+(1−𝛼)𝐴⁻¹.(2.4) For each 𝛼∈(0,1] and 𝛽≥0, we know that 𝐵^𝛽𝛼∶𝐻_𝑠→𝐻_𝑠 is bounded and has a bounded inverse (see [10, 11]). We also define the operator 𝐴_𝛼∶=𝐵^𝛼−1𝐴 with domain𝐷𝐴_𝛼=𝐷𝐷(𝐴𝐴)if𝛼>0,₀=𝐻₄.(2.5) By definition, it is clear that 𝐷(𝐴^𝛼𝑠/2)=𝐻_𝑠 in case 𝛼>0.

Lemma 2.1. For 𝛼>0, there exist 𝑀₁,𝑀₂, and 𝑀₃ such that 𝛼𝛼^1/2|𝑣|≤|𝑣|_𝐵𝛼≤𝑀^11/2|𝑣|,𝑣∈𝐻,(2.6)_1/2𝜆|𝑣|₁≤|𝑣|_1,𝐵𝛼≤𝑀^21/2|𝑣|₁,𝑣∈𝐻₁,(2.7)₁𝛼𝜆₁+1−𝛼^1/2|𝑣|≤|𝑣|_𝐵^𝛼−1≤𝑀^31/2|𝑣|,𝑣∈𝐻,(2.8) where |𝑣||_1,𝐵𝛼𝑣|_𝐵𝛼∶=𝑣,𝐵_𝛼𝑣^1/2,|𝑣|∶=_𝐵𝛼−1𝐴^1/2𝑣,𝐵_𝛼𝐴^1/2𝑣^1/2,∶=𝑣,𝐵^𝛼−1𝑣^1/2.(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 𝐵^𝛼−1/2 is bounded, there exists 𝑀₃≥0, such that |𝐵^𝛼−1/2|²≤𝑀₃. Then, for any 𝑣∈𝐻, we have 𝑣,𝐵^𝛼−1𝑣=𝐵^𝛼−1/2𝑣,𝐵^𝛼−1/2𝑣=||𝐵^𝛼−1/2𝑢||²≤𝑀₃|𝑣|²,(2.10) which completes the right part of (2.8).
Now, we proof the left part of (2.8) let 0<𝜆₁≤𝜆₂≤⋯≤𝜆_𝑘≤⋯(2.11) denote the eigenvalues of 𝐴, repeated with the respective multiplicity, and the corresponding unit eigenvector is denoted by {𝑤_𝑘}^∞𝑘=1, which forms an orthonormal basis for 𝐻. We have 𝑤_𝑘,𝐵^𝛼−1𝑤_𝑘=𝛼𝜆_𝑘+𝜆^𝑘1−𝛼≥𝛼𝜆₁+𝜆¹1−𝛼,𝑘∈ℤ⁺.(2.12) Since 𝑣∈𝐻, there exist {𝑏_𝑘}^∞𝑘=1⊂ℝ, such that 𝑣=∑^+∞𝑘=1𝑏_𝑘𝑤_𝑘. Consequently, 𝑣,𝐵^𝛼−1𝑣=^+∞_𝑘=1𝑏_𝑘𝑤_𝑘,𝐵^𝛼−1+∞_𝑘=1𝑏_𝑘𝑤_𝑘=^+∞_𝑘=1=_+∞𝑏_𝑘𝑤_𝑘,𝐵^𝛼−1𝑏_𝑘𝑤_𝑘_𝑘=1𝜆_𝑘𝛼𝜆_𝑘+1−𝛼𝑏^2𝑘≥𝜆¹𝛼𝜆₁+1−𝛼^+∞_𝑘=1𝑏^2𝑘=𝜆¹𝛼𝜆₁+1−𝛼|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,ℱ,𝐏), is a two-side in time Wiener process on 𝐻 which is given by the expansions 𝑊(𝑡)=^∞_𝑘=0√𝛼_𝑘𝛽_𝑘(𝑡)𝑤_𝑘,(2.14) where {𝑤_𝑘}^∞𝑘=1 is a basis of 𝐻 consisting of unit eigenvectors of 𝐴, {𝛼_𝑘}^∞𝑘=1 is a bounded sequence of nonnegative numbers, and 𝛽_𝑘(𝑡)=1√𝛼_𝑘𝑊(𝑡),𝑤_𝑘,𝑘∈ℕ(2.15) is a sequence of mutually independent real valued standard Brownian motions in a fixed probability space (Ω,ℱ,𝐏) adapted to a filtration {ℱ_𝑡}_𝑡≥0.

For convenience, we will define the covariance operator 𝑄 on 𝐻 as follows: 𝑄𝑤_𝑘=𝛼_𝑘𝑤_𝑘,𝑘∈ℕ.(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[𝐵^{𝛼−1−𝛿}𝐴^−2+𝛿𝑄]<∞(forsome0<𝛿≤1), (Q1*)Tr[𝐵^𝛼−2𝐴⁻¹𝑄]<∞,and Tr[𝐵^𝛼−2𝐴⁻²𝑄]≤2𝐷,(Q2)Tr[𝐵^{𝛼−1−𝛿}𝐴^−1+𝛿𝑄]<∞(for some0<𝛿≤1),Tr[𝐵^𝛼−2𝑄]<∞, and Tr[𝐵^𝛼−2𝐴⁻²𝑄]≤2𝐷,(Q2*)Tr[𝐵^{𝛼−1−𝛿}𝐴^−1+𝛿𝑄]<∞,Tr[𝐵^𝛼−2𝐴^𝜎𝑄]<∞ (for some 0<𝛿≤1 and 𝜎>0), and Tr[𝐵^𝛼−2𝐴⁻²𝑄]≤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=a2n,ypositiveinteger,iiff𝑛𝑛==31.or2,(2.18)

Under the above assumptions on the noise, we can now put the original problem (1.1)-(1.2) in an abstract form 𝑑𝑢+𝐴_𝛼𝑢+𝐵^𝛼−1𝑓(𝑢)𝑑𝑡=𝐵^𝛼−1𝐴⁻¹𝑑𝑊,(2.19) with which we will also associate the following initial condition: 𝑢𝑡₀=𝑢₀.(2.20) Note that since 𝐵^𝛼−1 is bounded from 𝐻_𝑠 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 ℝ. We say that a stochastic process 𝑢(𝑡,𝜔;𝑡₀,𝑢₀) is a mild solution of the system (2.19)-(2.20) in 𝐻_𝑠, if 𝑢⋅,𝜔;𝑡₀,𝑢₀∈𝐶𝐼;𝐻_𝑠,𝐏-a.s.𝜔∈Ω,(2.21) moreover, it satisfies in 𝐻_𝑠 the following integral equation: 𝑢𝑡,𝜔;𝑡₀,𝑢₀=𝑒^{−𝐴𝛼(𝑡−𝑡0)}𝑣₀−^𝑡𝑡₀𝑒^{−𝐴𝛼(𝑡−𝑠)}𝐵^𝛼−1𝑓(𝑢)−𝛽𝑊_𝐴(𝑠)𝑑𝑠+𝑊_𝐴(𝑡),𝐏-a.s.𝜔∈Ω,(2.22) where 𝑊_𝐴(𝑡) is called stochastic convolution (see Section 3 for details), 𝛽 is a positive constant chosen in Section 3 and 𝑣₀=𝑢₀−𝑊_𝐴(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 𝑢₀∈𝐻₂, 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 𝑢₀∈𝐻₁, 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 𝐵⊂𝐻₂.
(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 𝒜_𝛼(𝜔)⊂𝐻₁ at time 0, which pullback attracts every bounded deterministic set 𝐵⊂𝐻₁.

3. Stochastic Convolution

Let 𝑊_𝐴(𝑡) be the unique solution of linear equation 𝑑𝑢+𝐴_𝛼+𝛽𝑢𝑑𝑡=𝐵^𝛼−1𝐴⁻¹𝑑𝑊,(3.1) where 𝛽 is a positive constant to be further determined. Then, 𝑊_𝐴(𝑡) is an ergodic and stationary process [9, 12] called the stochastic convolution. Moreover, 𝑊_𝐴(𝑡)=^𝑡−∞𝑒^{−(𝑡−𝑠)(𝐴𝛼+𝛽)}𝐵^𝛼−1𝐴⁻¹𝑑𝑊(𝑠).(3.2)

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

Lemma 3.1. Assume that (𝐐𝟏) holds. Then, ∇𝑊_𝐴(𝑡) has a version which is 𝛾-Hölder continuous with respect to (𝑡,𝑥)∈ℝ×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: 𝑤_𝑘(𝑥)=2𝜋^3/2cos𝑘₁𝑥₁cos𝑘₂𝑥₂cos𝑘₃𝑥₃,𝑥=𝑥₁,𝑥₂,𝑥₃∈ℝ³,(3.3) with corresponding eigenvalues 𝜇_𝑘=𝑘²¹+𝑘²²+𝑘²³=||𝑘||²,𝑘∈ℤ⁺³,(3.4) where 𝑘=(𝑘₁,𝑘₂,𝑘₃) varies in (ℤ⁺)³. Using (2.14), we find that 𝑊_𝐴(𝑡,𝑥)=_{𝑘∈(ℤ+)3}√𝛼_𝑘^𝑡−∞𝑒^{−(𝑡−𝑠)(𝜂𝑘+𝛽)}1𝛼𝜇_𝑘+1−𝛼𝑑𝛽^𝑘(𝑠)𝑤_𝑘(𝑥),(3.5) where 𝜂_𝑘=𝜇^2𝑘/(𝛼𝜇_𝑘+1−𝛼), and hence, ∇𝑊_𝐴=(𝑡,𝑥)−∇𝑊_𝐴(𝑡,𝑦)_{𝑘∈(ℤ+)3}√𝛼_𝑘^𝑡−∞𝐄||∇𝑊_𝐴𝑒^{−(𝑡−𝑠)(𝜂𝑘+𝛽)(1/(𝛼𝜇𝑘+1−𝛼))}𝑑𝛽_𝑘(𝑠)∇𝑤_𝑘(𝑥)−∇𝑤_𝑘(𝑦),≤(𝑡,𝑥_{ℤ𝑘∈+3})−∇𝑊_𝐴(𝑡,𝑦)||²𝛼_𝑘𝛼𝜇_𝑘+1−𝛼²^𝑡−∞𝑒^{−2(𝑡−𝑠)(𝜂𝑘+𝛽)}𝑑𝑠||∇𝑤_𝑘(𝑥)−∇𝑤_𝑘(𝑦)||².(3.6)
For any 𝛾∈[0,1], one trivially verifies that there is a constant 𝑐_𝛾>0 independent of 𝑘 such that for any 𝑘∈(ℤ⁺)³ and 𝑥,𝑦∈𝐺||∇𝑤_𝑘(𝑥)−∇𝑤_𝑘(𝑦)||≤𝑐_𝛾𝜇^{𝑘(1+𝛾)/2}||𝑥−𝑦||^𝛾.(3.7) Thus, we have 𝐄||∇𝑊_𝐴(𝑡,𝑥)−∇𝑊_𝐴(𝑡,𝑦)||²≤𝑐^2𝛾2||𝑥−𝑦||^2𝛾_{ℤ𝑘∈+3}𝛼_𝑘𝛼𝜇_𝑘+1−𝛼²𝜂^𝑘−1𝜇^𝑘1+𝛾=𝑐^2𝛾2||𝑥−𝑦||^2𝛾_{ℤ𝑘∈+3}𝛼_𝑘𝛼𝜇_𝑘+1−𝛼²𝛼𝜇_𝑘+1−𝛼𝜇^2𝑘𝜇^𝑘1+𝛾=𝑐^2𝛾2||𝑥−𝑦||^2𝛾_{ℤ𝑘∈+3}𝛼_𝑘𝜇_𝑘𝛼𝜇_𝑘+1−𝛼𝜇^{𝑘−2+𝛾}.(3.8)
Now, let 𝑡,𝑠∈ℝ. We may assume that 𝑡≥𝑠. Then, 𝐄||∇=𝑊_𝐴(𝑡,𝑥_{ℤ𝑘∈+3})−∇𝑊_𝐴(𝑠,𝑥)||²𝛼_𝑘𝛼𝜇_𝑘+1−𝛼²×^𝑡𝑠𝑒^{−2(𝜂𝑘+𝛽)(𝑡−𝜎)}𝑑𝜎+^𝑠−∞𝑒^{−(𝜂𝑘+𝛽)(𝑡−𝜎)}−𝑒^{−(𝜂𝑘+𝛽)(𝑠−𝜎)}²𝑑𝜎||∇𝑤_𝑘(𝑥)||²=_{ℤ𝑘∈+3}𝛼_𝑘𝛼𝜇_𝑘+1−𝛼²12𝜂_𝑘+𝛽1−𝑒^{−2(𝜂𝑘+𝛽)(𝑡−𝑠)}⋅||∇𝑤_𝑘(𝑥)||².(3.9) Let 0≤𝛾≤1/2, and let 𝑐^𝛾=sup_{𝑟1,𝑟2≥0}||𝑒^−𝑟1−𝑒^−𝑟2||||𝑟₁−𝑟₂||^2𝛾.(3.10) Since the function g(𝑟)=𝑒^−𝑟 is a Lipschitzoneon [0,∞), we always have 𝑐^𝛾<∞. Observe that 𝐄||∇≤𝑊4^𝐴_𝛾(𝑡,𝑥)−∇𝑊_𝐴(𝑠,𝑥)||²𝜋³𝑐^𝛾|𝑡−𝑠|^2𝛾_𝑘∈_(ℤ+)3𝛼_𝑘𝛼𝜇_𝑘+1−𝛼²≤4_𝛾𝜂_𝑘+𝛽^2𝛾−1𝜇_𝑘.𝜋³𝑐^𝛾|𝑡−𝑠|^2𝛾_𝑘∈_(ℤ+)3𝛼_𝑘𝛼𝜇_𝑘+1−𝛼²𝜂^{𝑘2𝛾−1}𝜇_𝑘=4^𝛾𝜋³𝑐^𝛾|𝑡−𝑠|^2𝛾_𝑘∈_(ℤ+)3𝛼_𝑘𝛼𝜇_𝑘+𝜇^𝑘1−𝛼^2𝛾+1𝜇^{𝑘−2+2𝛾}.(3.11) By (𝐐𝟏), we know that Tr[𝐵^{𝛼−1−𝛿}𝐴^−2+𝛿𝑄]<∞ for some 0<𝛿≤1. Therefore, by (3.8) and (3.11), one deduces that there exists a constant 𝑐^𝛾>0 such that 𝐄||∇𝑊_𝐴(𝑡,𝑥)−∇𝑊_𝐴(𝑠,𝑦)||²≤𝑐^𝛾||𝑥−𝑦||²+|𝑡−𝑠|²^𝛾,∀(𝑡,𝑥),(𝑠,𝑦)∈ℝ×𝐺.(3.12) As 𝑊_𝐴(𝑡,𝑥)−𝑊_𝐴(𝑠,𝑦) is a Gaussian process, we find that for each 𝑚∈ℤ⁺, there is a constant 𝑐^𝑚𝛾>0 such that 𝐄||∇𝑊_𝐴(𝑡,𝑥)−∇𝑊_𝐴(𝑠,𝑦)||^2𝑚≤𝑐^𝑚𝛾||𝑥−𝑦||²+|𝑡−𝑠|²^𝑚𝛾.(3.13) Now, thanks to the well-known Kolmogorov test, one concludes that 𝑊_𝐴(𝑡,𝑥) is (𝛾−2/𝑚)-Hölder continuous in (𝑡,𝑥). Because 𝛾∈[0,1/2] and 𝑚∈ℤ⁺ 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 𝛽≥𝛽₀, 𝐄||𝑊_𝐴(𝑡)||²²≤𝑀.(3.14)

Proof. 𝐄||Δ𝑊_𝐴(𝑡)||²=𝐄⎛⎜⎜⎝_{𝑘∈(ℤ+)3}√𝛼_𝑘^𝑡−∞𝑒^{−(𝜂𝑘+𝛽)(𝑡−𝑠)}1𝛼𝜇_𝑘+1−𝛼𝑑𝛽^𝑘(𝑠)Δ𝑤_𝑘(𝑥)⎞⎟⎟⎠²=_{𝑘∈(ℤ+)3}𝛼_𝑘𝛼𝜇_𝑘+1−𝛼²^𝑡−∞𝑒^{−2(𝜂𝑘+𝛽)(𝑡−𝑠)}𝑑𝑠||Δ𝑤_𝑘(𝑥)||²≤_{ℤ𝑘∈+3}𝛼_𝑘𝛼𝜇_𝑘+1−𝛼²12𝜂_𝑘+𝛽||Δ𝑤^𝑘(𝑥)||²≤12𝜂₁+𝛽_{𝑘∈(ℤ+)3}𝛼_𝑘𝛼𝜇_𝑘+𝜇^𝑘1−𝛼².(3.15) Since Tr[𝐵^𝛼−2𝑄]<∞, one can now easily choose a 𝛽 large enough so that 𝐄(|Δ𝑊_𝐴(𝑡)|²)≤𝑀, and the proof is complete.

Similarly, we can verify the following basic fact.

Lemma 3.3. Assume (𝐐𝟐) holds. Then, Δ𝑊_𝐴 has a version which is 𝛾-Hölder continuous with respect to (𝑡,𝑥)∈ℝ×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 𝛽≥𝛽₀, 𝐄||𝑊_𝐴(𝑡)||^22+𝜎≤𝑀.(3.16)

4. Stochastic dissipativeness in 𝐻₁

It is well known that in the deterministic case without forcing terms, 𝐽(𝑢)=12||∇𝑢||²+_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 𝑑𝐽(𝑢)=𝐽_𝑢(𝑢),𝑑𝑢+12Tr𝐽^𝑢𝑢=𝐽(𝑢)𝐵^𝛼−2𝐴⁻²𝑄𝑑𝑡_𝑢(𝑢),𝐵^𝛼−1𝐴⁻¹𝑑𝑊−𝐽_𝑢(𝑢),𝐵^𝛼−1𝐴𝑢+𝐵^𝛼−1𝑓(𝑢)𝑑𝑡+12Tr𝐽^𝑢𝑢(𝑢)𝐵^𝛼−2𝐴⁻²𝑄𝑑𝑡,(4.2) where 𝐽_𝑢,𝐽_𝑢𝑢 denote, respectively, the first and second derivative of 𝐽. Since 𝐽_𝑢(𝑢)=𝐴𝑢+𝑓(𝑢),(4.3) there exists 𝐶₁>0 such that for 𝛼=0, 𝐽_𝑢(𝑢),𝐵^𝛼−1𝐴𝑢+𝐵^𝛼−1𝑓(𝑢)=||𝐴𝑢+𝑓(𝑢)||²¹≥𝜆²¹||𝐴𝑢+𝑓(𝑢)||²⁻¹==𝜆𝜆²¹₂₁𝐴𝑢+𝑓(𝑢),𝑢+𝐴⁻¹𝑓(𝑢)≥𝑑𝜆₂₁|𝑢|²¹+||𝑓(𝑢)||²⁻¹+2(𝑓(𝑢),𝑢)|𝑢|²¹+_𝐺𝐹(𝑢)𝑑𝑥−𝐶₁=𝑑𝜆²¹𝐽(𝑢)−𝐶₁,(4.4) where 𝑑=min{1,4𝑝𝑎_2𝑝−1}. And for 0<𝛼≤1, 𝐽_𝑢(𝑢),𝐵^𝛼−1𝐴𝑢+𝐵^𝛼−1𝑓(𝑢)==||𝐴𝐴𝑢𝑢++𝑓𝑓((𝑢𝑢))||,𝐵^𝛼−1_{2𝐵𝛼−1}𝐴𝑢+𝐵^𝛼−1𝑓(𝑢)≥𝜆²¹𝛼𝜆₁+1−𝛼||𝐴𝑢+𝑓(𝑢)||²⁻¹≥𝑑𝜆²¹𝛼𝜆₁+1−𝛼𝐽(𝑢)−𝐶¹,(4.5) where we have used (2.8). Simple computations show that 𝐽_𝑢𝑢(𝑢)=𝐴+𝑓^(𝑢),(4.6) and hence, Tr𝐽_𝑢𝑢(𝑢)𝐵^𝛼−2𝐴⁻²𝑄=Tr𝐴𝐵^𝛼−2𝐴⁻²𝑄+^∞_𝑖=1𝐷^𝑖_𝐺=Tr𝐵_𝛼−2𝑓^(𝑢)𝑤^2𝑖𝑑𝑥𝐴⁻¹𝑄+^∞_𝑖=1𝐷^𝑖_𝐺𝑓^(𝑢)𝑤^2𝑖𝑑𝑥,(4.7) where {𝑤_𝑖}^∞𝑖=1 is the orthonormal basis of 𝐻 as in (2.14), and 𝐷_𝑖=𝛼_𝑖/(𝛼𝜆_𝑖+1−𝛼)².

We infer from (3.3) that ||𝑤_𝑖||_𝐿∞≤𝐶₂,(4.8) where 𝐶₂>0 depends only on 𝐺. Therefore, ||||_𝐺𝑓^(𝑢)𝑤^2𝑖𝑑𝑥||||≤𝐶²²_𝐺||𝑓^(𝑢)||𝑑𝑥.(4.9) Set 𝐶₃ such that ||𝑓^(𝑠)||≤2(2𝑝−1)𝑎_2𝑝−1𝑠^2𝑝−2+𝐶₃,𝑠∈ℝ,(4.10) then||||_𝐺𝑓^(𝑢)𝑤^2𝑖𝑑𝑥||||≤𝐶²²2(2𝑝−1)𝑎^2𝑝−1_𝐺≤1𝑢^2𝑝−2𝑑𝑥+𝐶₃||𝐺||4𝑝𝑎^2𝑝−1_𝐺𝑢^2𝑝𝑑𝑥+𝐶₄,(4.11) where 𝐶₄ depends on 𝑓, 𝑝, and 𝐺. Let 𝐶₅ satisfy 𝐹(𝑠)≥14𝑝𝑎^2𝑝−1𝑠^2𝑝−𝐶⁵||𝐺||,𝑠∈ℝ,(4.12) then ||||_𝐺𝑓^(𝑢)𝑤^2𝑖𝑑𝑥||||≤𝐽(𝑢)+𝐶₄+𝐶₅.(4.13) Finally, Tr𝐽_𝑢𝑢(𝑢)𝐵^𝛼−2𝐴⁻²𝑄≤Tr𝐵^𝛼−2𝐴⁻¹𝑄+Tr𝐵^𝛼−2𝐴⁻²𝑄𝐽(𝑢)+𝐶₄+𝐶₅.(4.14) Since 𝐄𝐽_𝑢(𝑢),𝐵^𝛼−1𝐴⁻¹𝑑𝑊=0,(4.15) we have from (4.2) that 𝑑𝑑𝑡𝐄(𝐽(𝑢))=𝐄𝐽^𝑢(𝑢),−𝐵^𝛼−1𝐴(𝑢)−𝐵^𝛼−1(𝑢)+12𝐄Tr𝐽^𝑢𝑢(𝑢)𝐵^𝛼−2𝐴⁻²𝑄.(4.16) Further, by (4.4), (4.5) and (4.14), it holds that𝑑𝑑𝑡≤𝐄−(𝐽𝐷(𝑢)−)12Tr𝐵^𝛼−2𝐴⁻²𝑄𝐄(𝐽(𝑢))+Tr𝐵^𝛼−2𝐴⁻¹𝑄+Tr𝐵^𝛼−2𝐴⁻²𝑄𝐶₄+𝐶₅+𝐶₁,(4.17) where 𝐷=min{𝑑𝜆²¹,𝑑𝜆²¹/(𝛼𝜆₁+1−𝛼)}. 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 Tr𝐵^𝛼−2𝐴⁻¹𝑄<+∞,Tr𝐵^𝛼−2𝐴⁻²𝑄≤2𝐷,(4.18) and let 𝑢(𝑡) be the mild solution to (2.19)-(2.20). Then, 𝐄(𝐽(𝑢(𝑡)))≤𝐄𝐽𝑢₀+𝐶_𝑄,𝑡∈𝑡₀,∞,(4.19) where 𝐶_𝑄=Tr𝐵^𝛼−2𝐴⁻¹𝑄+Tr𝐵^𝛼−2𝐴⁻²𝑄𝐶₄+𝐶₅+𝐶₁𝐷−(1/2)Tr𝐵^𝛼−2𝐴⁻²𝑄.(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 Tr𝐵^𝛼−2𝐴⁻¹𝑄<+∞,Tr𝐵^𝛼−2𝐴⁻²𝑄≤2𝐷.(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)||²¹≤Ψ𝐄||𝑢⁰||²¹,∀𝑡∈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)−𝑊_𝐴(𝑡).(5.1)

We observe that 𝑣(𝑡) satisfies the following system: 𝑑𝑣𝑑𝑡+𝐴^𝛼𝑣−𝑡𝛽𝑣+𝐵^𝛼−1𝑓𝑣+𝑊_𝐴=0,₀=𝑢₀−𝑊_𝐴𝑡₀.(5.2) Let 𝐺(𝑣,𝑡)=−𝐵^𝛼−1𝑓𝑣+𝑊_𝐴+𝛽𝑊_𝐴,𝑣₀=𝑢₀−𝑊_𝐴𝑡₀.(5.3) Then, (5.2) reads 𝑑𝑣𝑑𝑡+𝐴^𝛼𝑣𝑡𝑣=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 ℝ. We say that a stochastic process 𝑣(𝑡,𝜔;𝑡₀,𝑣₀) is a mild solution of the system (5.4) in 𝐻_𝑠, if 𝑣⋅,𝜔;𝑡₀,𝑣₀∈𝐶𝐼;𝐻_𝑠,𝐏-a.s.𝜔∈Ω,(5.5) and satisfies in 𝐻_𝑠 the following integral equation: 𝑣𝑡,𝜔;𝑡₀,𝑣₀=𝑒^{−𝐴𝛼(𝑡−𝑡0)}𝑣₀−^𝑡𝑡₀𝑒^{−𝐴𝛼(𝑡−𝑠)}𝐵^𝛼−1𝑓(𝑢)−𝛽𝑊_𝐴(𝑠)𝑑𝑠,𝐏-a.s.𝜔∈Ω.(5.6)

Theorem 5.2. Let 𝛼=0. Suppose that the Hypothesis (Q2) is satisfied.
Then, for every 𝑢₀∈𝐻₂, there is a unique globally defined mild solution 𝑣(𝑡,𝜔;𝑡₀,𝑣₀) of (5.4) in 𝐻₂ with 𝑣𝑡,𝜔;𝑡₀,𝑣₀∈𝐶𝑡₀,∞;𝐻₂∩𝐶^{0,1−𝑟loc}𝑡₀,;𝐻_4𝑟∩𝐶𝑡₀,∞;𝐻₄,(5.7) for all 0≤𝑟<1.

Proof. We only consider the case where 𝑛=3. First, it is easy to verify that 𝐏-a.s. 𝐺(𝑣,𝑡)∈𝐶_Lip;𝛾𝐻₂×𝑡₀,∞,𝐻.(5.8) Indeed, by Lemma 3.3, we see that 𝑊_𝐴(𝑡)∈𝐻₂ is 𝛾-Hölder continuous with respect to 𝑡∈ℝ   𝐏-a.s. Recall that 𝑓 is a polynomial of degree 2𝑝−1 with 𝑝=2 (in case 𝑛=3). One deduces that there exist 𝐶₁,𝐶₂(𝜔)>0 such that ||𝐺𝑣₁,𝑡₁−𝐺𝑣₂,𝑡₂||≤≤𝐶𝐶₁₂||𝑣₁−𝑣₂||₂+||𝑊_𝐴𝑡₁−𝑊_𝐴𝑡₂||₂(𝜔)||𝑣₁−𝑣₂||₂+||𝑡₁−𝑡₂||^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. 𝑣𝑡,𝑣𝜔;𝑡𝑡,₀𝜔;𝑡₀,𝑣₀=𝑒^{−𝐴2(𝑡−𝑡0)}𝑣₀−^𝑡𝑡₀𝑒^{−𝐴2(𝑡−𝑠)}𝐴𝑓(𝑢(𝑠))−𝛽𝑊_𝐴(𝑠)𝑑𝑠,,𝑣₀∈𝐶𝑡₀,𝑇;𝐻₂∩𝐶^{0,1−𝑟loc}𝑡₀,𝑇;𝐻_4𝑟∩𝐶𝑡₀,𝑇;𝐻₄,(5.10) for all 0≤𝑟<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)−𝛽𝑊_𝐴=0,𝑣𝑡₀=𝑣₀.(5.11) In what follows, we show 𝑇=∞, thus proving the theorem.
Simple computations yields ||Δ𝑓(𝑢)||≤||𝑓^(𝑢)||_𝐿∞||Δ𝑢||+||𝑓^(𝑢)||_𝐿∞||∇𝑢||^2𝐿₄.(5.12) Since 𝑓 is a polynomial of degree 3, there exist 𝜅₁ and 𝜅₂ such that ||𝑓^(𝑠)||≤𝜅₁1+|𝑠|²,||𝑓^(𝑠)||≤𝜅₂(1+|𝑠|),∀𝑠∈ℝ.(5.13) Therefore, ||𝑓^(𝑢)||_𝐿∞||Δ𝑢||≤≤2𝜅𝜅₁₁1+|𝑢|^2𝐿∞||Δ𝑢||1+|𝑣|^2𝐿∞+||𝑊_𝐴||^2𝐿_∞||Δ𝑣||+||Δ𝑊^𝐴||.(5.14) By the Nirenberg-Gagiardo inequality, there exist 𝐶₃,𝐶₄,𝐶₅>0 such that |𝑢|_2𝐿∞|𝑢|^2𝐿_∞≤𝐶₃||Δ𝑢||²,𝑢∈𝐻₂,||Δ𝑢||≤≤𝐶𝐶₄₅||Δ²𝑢||^1/3|𝑢|^𝐿5/3₆,𝑢∈𝐻₄,||Δ²𝑢||^1/2||∇𝑢||^1/2,𝑢∈𝐻₄.(5.15) Hence, ||𝑓^(𝑢)||_𝐿∞||Δ𝑢||≤2𝜅₁1+|𝑣|^2𝐿∞+||𝑊_𝐴||^2𝐿_∞||Δ𝑣||+||Δ𝑊^𝐴≤2𝜅₁||1+𝐶⁴||Δ²𝑣||^1/3|𝑣|^𝐿5/3₆+𝐶₃||Δ𝑊_𝐴||²𝐶⁵||∇𝑣||^1/2||Δ²𝑣||^1/2+||Δ𝑊_𝐴||.(5.16) By (Q2) and Lemma 3.2, we know that for 𝐏-a.s. 𝜔∈Ω, there exists an 𝑅₁(𝜔)>0 such that |Δ𝑊_𝐴(𝑡)|≤𝑅₁(𝜔) (for all 𝑡∈ℝ). 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. 𝜔∈Ω, there exist 𝐶₆(𝜔),𝐶₇(𝜔)>0 such that |𝑣|^𝐿5/3₆≤𝐶₆(𝜔),||∇𝑣||^1/2≤𝐶₇(𝜔),(5.17) where the continuous imbedding 𝐻¹↪𝐿⁶ is used. Consequently, we have ||𝑓^(𝑢)||_𝐿∞||Δ𝑢||≤𝐶₈(𝜔)1+||Δ²𝑣||^1/3+𝑅₁(𝜔)||Δ²𝑣||^1/2+𝑅₁(𝜔),𝐏-a.s.𝜔∈Ω.(5.18)
Similarly for 𝐏-a.s. 𝜔∈Ω, one easily deduces that there exists 𝐶₉(𝜔)>0 such that ||𝑓^(𝑢)||_𝐿∞||Δ𝑢||≤𝐶₉(𝜔)1+||Δ²𝑣||^1/6+𝑅₁(𝜔)||Δ²𝑣||^1/4+𝑅₁(𝜔).(5.19) It then follows from (5.12) that for 𝐏-a.s. 𝜔∈Ω, ||Δ𝑓(𝑢)||≤+𝐶₈𝐶(₉𝜔)1+||Δ²𝑣||^1/3+𝑅₁(𝜔)||Δ²𝑣||^1/2+𝑅₁(𝜔)≤𝐶₁₀(𝜔)1+𝐿₃||Δ²𝑣||^1/6+𝑅₁(𝜔)||Δ²𝑣||^1/4+𝑅₁(𝜔)(𝜔)1+||Δ²𝑣||^5/6.(5.20)
Now, taking the 𝐿² inner-product of equation (5.11) with Δ²𝑣, one obtains 12𝑑𝑑𝑡||Δ𝑣||²+||Δ²𝑣||²≤≤1||||_𝐺Δ𝑓(𝑢)Δ²𝑣𝑑𝑥||||+𝛽||||_𝐺𝑊_𝐴Δ²𝑣𝑑𝑥||||4||Δ²𝑣||²+||Δ𝑓(𝑢)||²+14||Δ²𝑣||²+𝛽²||𝑊_𝐴||²≤12||Δ²𝑣||²+||Δ𝑓(𝑢)||²+𝛽²𝜆¹⁻²||Δ𝑊_𝐴||².(5.21) By (5.20), we deduce that 𝐏-a.s. 𝑑𝑑𝑡||Δ𝑣||²+||Δ²𝑣||²≤𝐶₁₁(𝜔)1+||Δ²𝑣||^5/3.(5.22) Furthermore, by Young's inequality and |Δ²𝑣|²≥𝜆²¹|Δ𝑣|², we know that there exists 𝐶₁₂(𝜔)>0 such that 𝐏-a.s. 𝑑𝑑𝑡||Δ𝑣||²≤−𝜆²¹2||Δ𝑣||²+𝐶₁₂(𝜔).(5.23) Applying the gronwall lemma on (5.23), one gets ||Δ𝑣||²≤2𝐶¹²(𝜔)𝜆²¹,𝐏-a.s.𝜔∈Ω.(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 𝑢₀∈𝐻₂.

Theorem 5.3. Let 𝛼∈(0,1], and let Hypothesis (Q1) be satisfied. Then, for every 𝑢₀∈𝐻₁, there is a unique maximally defined mild solution 𝑣(𝑡,𝜔;𝑡₀,𝑣₀) of (5.4) in 𝐻₁ for all 𝑡∈[𝑡₀,∞) with 𝑣𝑡,𝜔;𝑡₀,𝑣₀∈𝐶𝑡₀,∞;𝐻₁∩𝐶^{0,1−𝑟loc}𝑡₀,∞;𝐻_2𝑟∩𝐶𝑡₀,∞;𝐻₂,(5.25) for 0≤𝑟<1.

Proof. As noted above, −𝐴_𝛼 is a positive selfadjoint linear operator on 𝐻 with compact resolvent. The negative operator −𝐴_𝛼 generate an analytic semigroup 𝑒^{−𝐴𝛼𝑡}. It is easy to verify by Lemma 3.1 that 𝐏-a.s. 𝐺(𝑣,𝑡)∈𝐶_Lip;𝛾𝐻₁×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 𝑣𝑡,𝜔;𝑡₀,𝑣₀∈𝐶𝑡₀,𝑇;𝐻₁∩𝐶^{0,1−𝑟loc}𝑡₀,𝑇;𝐻_2𝑟∩𝐶𝑡₀,𝑇;𝐻₂,(5.27) where 𝑡₀<𝑇=𝑇(𝑣₀)≤∞, and 0≤𝑟<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 𝐵_𝛼𝑑𝑣𝑑𝑡+𝐴𝑣=−𝑓𝑣+𝑊^𝐴+𝛽𝐵_𝛼𝑊_𝐴,𝑣𝑡₀=𝑣₀.(5.28) Multiplying (5.28) by 𝐴𝑣, one gets 12𝑑𝑑𝑡|𝑣|^21,𝐵𝛼+|𝑣|²²+𝑓𝑣+𝑊_𝐴,𝐴𝑣+𝛽𝐵_𝛼𝑊_𝐴,𝐴𝑣=0.(5.29) We observe that 𝑓𝑣+𝑊_𝐴,𝐴𝑣==_𝐺∇𝑓𝑣+𝑊_𝐴∇𝑣𝑑𝑥_𝐺𝑓^𝑣+𝑊_𝐴||∇𝑣||²𝑑𝑥+_𝐺𝑓^𝑣+𝑊_𝐴∇𝑊_𝐴∇𝑣𝑑𝑥.(5.30) We take 𝐶^1 and 𝐶^2 such that 𝑓^(𝑥)≥2𝑝−12𝑎^2𝑝−1||𝑓_𝑥^2𝑝−2−𝐶^1,(𝑥)||≤2(2𝑝−1)𝑎_2𝑝−1𝑥^2𝑝−2+𝐶′₂,(5.31) for all 𝑥∈ℝ. Then, 𝑓𝑣+𝑊_𝐴,𝐴𝑣≥−2𝑝22−(2𝑝1𝑎−1)_2𝑝−1𝑎_𝐺||𝑣_2𝑝−1+𝑊_𝐴||^2𝑝−2||∇𝑣||²𝑑𝑥−𝐶^1_𝐺||∇𝑣||²𝑑𝑥_𝐺||𝑣+𝑊_𝐴||^2𝑝−2||∇𝑣||||∇𝑊_𝐴||𝑑𝑥−𝐶^2_𝐺≥1||∇𝑣||||∇𝑊_𝐴||𝑑𝑥4(2𝑝−1)𝑎^2𝑝−1_𝐺||𝑣+𝑊_𝐴||^2𝑝−2||∇𝑣||²𝑑𝑥−2𝐶^1_𝐺−𝐶_3||∇𝑣||²𝑑𝑥_𝐺||∇𝑊_𝐴||^2𝑝𝑑𝑥+_𝐺||∇𝑊_𝐴||²𝑑𝑥,(5.32) where we have used Hölder's inequality, Young's inequality, and the appropriate imbeddings 𝐻¹(𝐺)↪𝐿^𝑟(𝐺) in dimension 𝑛=1𝑜𝑟2 and 3. We also know by (2.6) that there exists 𝛼≤𝐶_𝛼≤𝑀₁ such that 𝐵_𝛼𝑊_𝐴,𝐴𝑣≤𝐶_𝛼_𝐺||∇𝑊_𝐴||²𝑑𝑥+_𝐺||∇𝑣||²𝑑𝑥.(5.33) Combining the last two inequalities together, we deduce that there exists constants 𝐶^4,𝐶^5>0 such that 12𝑑𝑑𝑡|𝑣|^21,𝐵𝛼+|𝑣|²²+14(2𝑝−1)𝑎^2𝑝−1_𝐺≤2𝐶_4||𝑣+𝑊_𝐴||^2𝑝−2||∇𝑣||²𝑑𝑥_𝐺||∇𝑣||²𝑑𝑥+𝐶^5_𝐺||∇𝑊_𝐴||^2𝑝𝑑𝑥+_𝐺||∇𝑊_𝐴||²𝑑𝑥.(5.34) In view of (2.7), there exists 𝛼≤𝐶^𝛼≤𝑀₁ such that 12𝐶^𝛼𝑑𝑑𝑡|𝑣|²¹+|𝑣|²²+14(2𝑝−1)𝑎^2𝑝−1_𝐺≤2𝐶_4||𝑣+𝑊_𝐴||^2𝑝−2||∇𝑣||²𝑑𝑥|𝑣|²¹+𝐶^5_𝐺||∇𝑊_𝐴||^2𝑝𝑑𝑥+_𝐺||∇𝑊_𝐴||²𝑑𝑥.(5.35) Using the gronwall lemma on (5.35), the following inequality holds: |𝑣|²¹≤2𝑒^{4𝐶′4/𝐶′𝛼}𝐶′₄||𝑣₀||²¹+2𝑒^{4𝐶′4/𝐶′𝛼}^𝑡𝑡₀𝐶^5_𝐺||∇𝑊_𝐴(𝑠)||^2𝑝𝑑𝑥+_𝐺||∇𝑊_𝐴(𝑠)||²𝑑𝑥𝑑𝑠.(5.36) Lemma 3.1 guarantees that 𝐏-a.s. ^𝑡𝑡₀_𝐺||∇𝑊_𝐴(𝑠)||^2𝑝𝑑𝑥𝑑𝑠<+∞,^𝑡𝑡₀_𝐺||∇𝑊_𝐴(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 𝑢₀↦𝑢(𝑡,𝜔;𝑡₀,𝑢₀) defines a stochastic flow 𝑆_𝛼(𝑡,𝑠;𝜔),𝑆_𝛼(𝑡,𝑠;𝜔)𝑢₀=𝑢𝑡,𝜔;𝑠,𝑢₀,𝛼∈[0,1].(6.1) Notice that 𝐏-a.s. (i)𝑆_𝛼(𝑡,𝑠;𝜔)=𝑆_𝛼(𝑡,𝑟;𝜔)∘𝑆_𝛼(𝑟,𝑠;𝜔),forall𝑠≤𝑟≤𝑡,(ii)𝑆₀(𝑡,𝑠;𝜔) is continuous in 𝐻₂, and 𝑆_𝛼(𝑡,𝑠;𝜔) is continuous in 𝐻₁ for 0<𝛼≤1.

6.2. Compactness Properties of Stochastic Flow 𝑆_𝛼(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 𝐵⊂𝐻₂ bounded and each 𝑡₀<0, 𝑆₀(0,𝑡₀;𝜔)𝐵 is v relatively compact in 𝐻₂.
(ii) Under Assumption (𝐐𝟏^∗), the flow 𝑆_𝛼(𝑡,𝑠;𝜔), 0<𝛼≤1, is uniformly compact at time 0. More precisely, for all 𝐵⊂𝐻₁ bounded and each 𝑡₀<0, 𝑆_𝛼(0,𝑡₀;𝜔)𝐵 is 𝐏-a.s. relatively compact in 𝐻₁.

Proof. (i) Let 𝐵⊂𝐻₂ be a given bounded deterministic set. By Lemma 3.4, we know that for 𝐏-a.s. 𝜔∈Ω, there exists 𝑅₂(𝜔)>0, such that |𝑊_𝐴(𝑡)|_2+𝜎≤𝑅₂(𝜔), 𝑡∈ℝ. Define 𝐵=𝐵∪𝐵_2+𝜎(0,𝑅₂(𝜔)), where 𝐵_2+𝜎(0,𝑅₂(𝜔)) denotes the open ball centered at 0 with radius 𝑅₂(𝜔) in 𝐻_2+𝜎. Then, 𝐵⊂𝐻₂ is 𝐏-a.s. bounded, and 𝑆₀0,𝑡₀;𝜔𝐵⊂𝑒^𝐴2𝑡0𝑣₀−^0𝑡₀𝑒^𝐴2𝑠𝐺(𝑣(𝑠),𝑠)𝑑𝑠+𝑊_𝐴(0),𝑣₀∈𝐵⊂𝑁₁+𝑁₂+𝑁₃+𝑁₄,(6.2)𝐏-a.s., where 𝑁𝑁₁=𝑒^𝐴2𝑡0𝐵,₂𝑁₃=^0−𝛿𝑒^𝐴2𝑠𝐺(𝑣(𝑠),𝑠)𝑑𝑠,𝑣₀∈𝐵,=𝑒^−𝐴2𝛿^𝑡−𝛿₀𝑁₄𝑒^{𝐴2(𝑠+𝛿)}𝐺(𝑣(𝑠),𝑠)𝑑𝑠,𝑣₀∈𝐵,=𝐵_2+𝜎0,𝑅₂(𝜔),(6.3) and 𝛿 is an arbitrary constant satisfying 0<𝛿<−𝑡₀.

Since for 𝑡>0 fixed the operator 𝑒^−𝐴2𝑡 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 𝑁₂⊂𝐻₂.

For 𝑣₀∈𝐵, one has ||||^0−𝛿𝑒^{𝐴2(𝑠−𝑡0)}𝐺(𝑣(𝑠),𝑠)𝑑𝑠||||₂=||||^0−𝛿𝐴𝑒^{𝐴2(𝑠−𝑡0)}𝐺(𝑣(𝑠),𝑠)𝑑𝑠||||.(6.4) Since 𝐴² is a positive sectorial operator on 𝐻, there exists a constant 𝑀_𝐴>0 such that |||𝐴𝑒_−𝐴²_𝑡|||_ℒ(𝐻2)≤𝑀_𝐴𝑡^−1/2,∀𝑡≥0.(6.5) Recall that 𝐺(𝑣,𝑡)∈𝐶_Lip;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 ||||^0−𝛿𝑒^𝐴2𝑠𝐺(𝑣(𝑠),𝑠)𝑑𝑠||||₂≤𝐾₀(𝜔)𝑀_𝐴^0−𝛿(−𝑠)^1/2𝑑𝑠=12𝐾⁰(𝜔)𝑀_𝐴𝛿^1/2.(6.7) It follows that 𝜅𝑁₂≤diam_𝐻2𝑁₂≤𝐾₀𝑀_𝐴𝛿^1/2,(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 𝜅𝑆₀0,𝑡₀;𝜔𝐵≤𝜅𝑁₁+𝜅𝑁₂+𝜅𝑁₃+𝜅𝑁₄≤𝜅𝑁₂≤𝐾₀𝑀_𝐴𝛿^1/2.(6.9) Letting 𝛿→0, one immediately concludes that 𝐏-a.s. 𝜅(𝑆₀(0,𝑡₀;𝜔)𝐵)=0, hence 𝑆₀(0,𝑡₀;𝜔)𝐵 is relatively compact.

(ii) The proof of the compactness result for 𝑆_𝛼(𝑡,𝑠;𝜔) (0<𝛼≤1) is fully analogous, and is thus omitted.

6.3. The Random Attractors

Now, we show that the system 𝑆_𝛼(𝑡,𝑠;𝜔) possesses a random attractor 𝒜_𝛼(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 𝑡₀≤𝑡(𝜔), we can define an absorbing set for 𝑆₀(𝑡,𝑡₀;𝜔) at time 0 by 𝔅₀=𝑣∶||Δ𝑣||²≤2𝐶¹²(𝜔)𝜆²¹∪𝐵^2+𝜎0,𝑅₂(𝜔),(6.10) and for 𝑆_𝛼(𝑡,𝑠;𝜔) (0<𝛼≤1), for any 𝑡₀<0 we can define an absorbing set for 𝑆_𝛼(𝑡,𝑡₀;𝜔) at time 0 by 𝔅_𝛼=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

Appendix

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,ℱ,𝐏) be a probability space. We consider a family of mappings {𝑆(𝑡,𝑠;𝜔)}_{𝑡≥𝑠,𝜔∈Ω}∶𝑋⟶𝑋,(A.1) satisfying 𝐏-a.s. (i)𝑆(𝑡,𝑠;𝜔)=𝑆(𝑡,𝑟;𝜔)∘𝑆(𝑟,𝑠;𝜔),forall𝑠≤𝑟≤𝑡,(ii)𝑆(𝑡,𝑠;𝜔) is continuous in 𝑋, for all 𝑠≤𝑡.

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

Definition A.2. Given 𝑡∈ℝ and 𝜔∈Ω, we say that {𝑆(𝑡,𝑠;𝜔)}_{𝑡≥𝑠,𝜔∈Ω} is uniformly compact at time t if for all bounded set 𝐵⊂𝑋, there exist 𝑠_𝐵, such that 𝐏-a.s. _𝑠_≤𝑠𝐵𝑆(𝑡,𝑠;𝜔)𝐵(A.2) is relatively compact in 𝑋.

Definition A.3. Given 𝑡∈ℝ and 𝜔∈Ω, for any set 𝐵⊂𝑋, we define the random omega limit set of a bounded set 𝐵⊂𝑋 at time 𝑡 as Ω_𝐵(𝑡,𝜔)=_𝑇≤𝑡_𝑠≤𝑇𝑆(𝑡,𝑠;𝜔)𝐵.(A.3)

Definition A.4. Let (𝑋,𝑑) be a metric space, and let {𝑆(𝑡,𝑠;𝜔)}_{𝑡≥𝑠,𝜔∈Ω} 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 𝜏≥𝑠,(iii)for every bounded set 𝐵⊂𝑋, lim_{𝑡→−∞}𝑑(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 𝐵⊂𝑋 and {𝑆(𝑡,𝑠;𝑤)}_{𝑡≥𝑠,𝜔∈Ω} is uniformly compact at time 𝑡, then the RDS possesses a random attractor defined by 𝒜(𝑡,𝜔)=_𝐵_⊂𝑋Ω_𝐵(𝑡,𝜔).(A.4)

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

Acknowledgment

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