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

Authors: Xuewei Ju; Hongli Wang; Desheng Li; Jinqiao Duan
Journal: Abstract and Applied Analysis
 (2011)

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## 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.


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## Body

## 1. Introduction

This paper is devoted to the existence of mild solutions and global asymptotic behavior for the following stochastic viscous Cahn-Hilliard equation:(1.1)d((1-α)u-αΔu)+(Δ2u-Δf(u))dt=dW,(x,t)∈G×(t0,∞),
subjected to homogeneous Dirichlet boundary conditions  (1.2)u(x,t)=0,(x,t)∈∂G×[t0,∞),
in dimension n=1,2 or 3, where G=∏i=1n(0,Li) in Rn, and α∈[0,1] is a parameter, f is a polynomial of odd degree with a positive leading coefficient  (1.3)f(x)=∑k=12p-1akxk,a2p-1>0.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 caseα=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 Section2, 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)⋂H01(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:  (2.1)|u|s1≤λ1-(s2-s1)/2|u|s2,∀s1,s2∈R,s1<s2,∀u∈Hs2,(2.2)|u|σs1+(1-σ)s2≤|u|s1σ|u|s21-σ,σ∈[0,1],
where λ1 is the first eigenvalue of A.We can defineA-1:H→D(A) to be the Green’s operator for A. Thus, (2.3)v=A-1w⟺Av=w.
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 Bα:Hs→Hs, s∈ℝ defined by (2.4)Bα:=αI+(1-α)A-1.
For each α∈(0,1] and β≥0, we know that Bαβ:Hs→Hs is bounded and has a bounded inverse (see [10, 11]). We also define the operator Aα:=Bα-1A with domain(2.5)D(Aα)={D(A)ifα>0,D(A0)=H4.
By definition, it is clear that D(Aαs/2)=Hs in case α>0.Lemma 2.1.
Forα>0,  there exist M1,M2,  and M3 such that
(2.6)α1/2|v|≤|v|Bα≤M11/2|v|,v∈H,(2.7)α1/2|v|1≤|v|1,Bα≤M21/2|v|1,v∈H1,(2.8)(λ1αλ1+1-α)1/2|v|≤|v|Bα-1≤M31/2|v|,v∈H,
where
(2.9)|v|Bα:=(v,Bαv)1/2,|v|1,Bα:=(A1/2v,BαA1/2v)1/2,|v|Bα-1:=(v,Bα-1v)1/2.Proof.
Here, we only verify (2.8) is valid; the proofs of (2.6) and (2.7) can be found in [11]. Since Bα-1/2 is bounded, there exists M3≥0, such that |Bα-1/2|2≤M3. Then, for any v∈H, we have
(2.10)(v,Bα-1v)=(Bα-1/2v,Bα-1/2v)=|Bα-1/2u|2≤M3|v|2,
which completes the right part of (2.8).
Now, we proof the left part of (2.8) let
(2.11)0<λ1≤λ2≤⋯≤λk≤⋯
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
(2.12)(wk,Bα-1wk)=λkαλk+1-α≥λ1αλ1+1-α,k∈(Z+).
Since v∈H, there exist {bk}k=1∞⊂ℝ, such that v=∑k=1+∞bkwk. Consequently,
(2.13)(v,Bα-1v)=(∑k=1+∞bkwk,Bα-1∑k=1+∞bkwk)=∑k=1+∞(bkwk,Bα-1bkwk)=∑k=1+∞λkαλk+1-αbk2≥λ1αλ1+1-α∑k=1+∞bk2=λ1αλ1+1-α|v|2,
which finishes the proof.


### 2.2. Assumptions on the Noise

The stochastic processW(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  (2.14)W(t)=∑k=0∞αkβk(t)wk,
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 (2.15)βk(t)=1αk(W(t),wk),k∈N
is a sequence of mutually independent real valued standard Brownian motions in a fixed probability space (Ω,ℱ,P) adapted to a filtration {ℱt}t≥0.For convenience, we will define thecovariance operatorQ on H as follows: (2.16)Qwk=αkwk,k∈N.
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[Bα-1-δA-2+δQ]<∞(for some0<δ≤1),(Q1*)
Tr[Bα-2A-1Q]<∞,and Tr[Bα-2A-2Q]≤2D,(Q2)
Tr[Bα-1-δA-1+δQ]<∞(for  some0<δ≤1),Tr[Bα-2Q]<∞, and Tr[Bα-2A-2Q]≤2D,(Q2*)
Tr[Bα-1-δA-1+δQ]<∞,Tr[Bα-2AσQ]<∞ (for  some 0<δ≤1 and σ>0), and Tr[Bα-2A-2Q]≤2D,whereD is given in Section 4. It is obvious that (2.17)(Q2*)⟹(Q2),(Q1*)⟹(Q1).


### 2.3. Main Results

We will assume throughout the paper that the space dimensionn and the integer p in (1.3) satisfy the following  growth condition: (2.18)p={anypositiveinteger,ifn=1or2,2,ifn=3.Under the above assumptions on the noise, we can now put the original problem (1.1)-(1.2) in an abstract form  (2.19)du+(Aαu+Bα-1f(u))dt=Bα-1A-1dW,
with which we will also associate the following initial condition:  (2.20)u(t0)=u0.
Note that since Bα-1 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.
LetI:=[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 Hs,if
(2.21)u(⋅,ω;t0,u0)∈C(I;Hs),P-a.s.ω∈Ω,
moreover, it satisfies in Hs the following integral equation:
(2.22)u(t,ω;t0,u0)=e-Aα(t-t0)v0-∫t0te-Aα(t-s)(Bα-1f(u)-βWA(s))ds+WA(t),P-a.s.ω∈Ω,
where WA(t) is called stochastic convolution (see Section 3 for details), β is a positive constant chosen in Section 3 and v0=u0-WA(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) Letα=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) Letα∈(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 B⊂H1.


## 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)⋂H01(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:  (2.1)|u|s1≤λ1-(s2-s1)/2|u|s2,∀s1,s2∈R,s1<s2,∀u∈Hs2,(2.2)|u|σs1+(1-σ)s2≤|u|s1σ|u|s21-σ,σ∈[0,1],
where λ1 is the first eigenvalue of A.We can defineA-1:H→D(A) to be the Green’s operator for A. Thus, (2.3)v=A-1w⟺Av=w.
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 Bα:Hs→Hs, s∈ℝ defined by (2.4)Bα:=αI+(1-α)A-1.
For each α∈(0,1] and β≥0, we know that Bαβ:Hs→Hs is bounded and has a bounded inverse (see [10, 11]). We also define the operator Aα:=Bα-1A with domain(2.5)D(Aα)={D(A)ifα>0,D(A0)=H4.
By definition, it is clear that D(Aαs/2)=Hs in case α>0.Lemma 2.1.
Forα>0,  there exist M1,M2,  and M3 such that
(2.6)α1/2|v|≤|v|Bα≤M11/2|v|,v∈H,(2.7)α1/2|v|1≤|v|1,Bα≤M21/2|v|1,v∈H1,(2.8)(λ1αλ1+1-α)1/2|v|≤|v|Bα-1≤M31/2|v|,v∈H,
where
(2.9)|v|Bα:=(v,Bαv)1/2,|v|1,Bα:=(A1/2v,BαA1/2v)1/2,|v|Bα-1:=(v,Bα-1v)1/2.Proof.
Here, we only verify (2.8) is valid; the proofs of (2.6) and (2.7) can be found in [11]. Since Bα-1/2 is bounded, there exists M3≥0, such that |Bα-1/2|2≤M3. Then, for any v∈H, we have
(2.10)(v,Bα-1v)=(Bα-1/2v,Bα-1/2v)=|Bα-1/2u|2≤M3|v|2,
which completes the right part of (2.8).
Now, we proof the left part of (2.8) let
(2.11)0<λ1≤λ2≤⋯≤λk≤⋯
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
(2.12)(wk,Bα-1wk)=λkαλk+1-α≥λ1αλ1+1-α,k∈(Z+).
Since v∈H, there exist {bk}k=1∞⊂ℝ, such that v=∑k=1+∞bkwk. Consequently,
(2.13)(v,Bα-1v)=(∑k=1+∞bkwk,Bα-1∑k=1+∞bkwk)=∑k=1+∞(bkwk,Bα-1bkwk)=∑k=1+∞λkαλk+1-αbk2≥λ1αλ1+1-α∑k=1+∞bk2=λ1αλ1+1-α|v|2,
which finishes the proof.


## 2.2. Assumptions on the Noise

The stochastic processW(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  (2.14)W(t)=∑k=0∞αkβk(t)wk,
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 (2.15)βk(t)=1αk(W(t),wk),k∈N
is a sequence of mutually independent real valued standard Brownian motions in a fixed probability space (Ω,ℱ,P) adapted to a filtration {ℱt}t≥0.For convenience, we will define thecovariance operatorQ on H as follows: (2.16)Qwk=αkwk,k∈N.
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[Bα-1-δA-2+δQ]<∞(for some0<δ≤1),(Q1*)
Tr[Bα-2A-1Q]<∞,and Tr[Bα-2A-2Q]≤2D,(Q2)
Tr[Bα-1-δA-1+δQ]<∞(for  some0<δ≤1),Tr[Bα-2Q]<∞, and Tr[Bα-2A-2Q]≤2D,(Q2*)
Tr[Bα-1-δA-1+δQ]<∞,Tr[Bα-2AσQ]<∞ (for  some 0<δ≤1 and σ>0), and Tr[Bα-2A-2Q]≤2D,whereD is given in Section 4. It is obvious that (2.17)(Q2*)⟹(Q2),(Q1*)⟹(Q1).


## 2.3. Main Results

We will assume throughout the paper that the space dimensionn and the integer p in (1.3) satisfy the following  growth condition: (2.18)p={anypositiveinteger,ifn=1or2,2,ifn=3.Under the above assumptions on the noise, we can now put the original problem (1.1)-(1.2) in an abstract form  (2.19)du+(Aαu+Bα-1f(u))dt=Bα-1A-1dW,
with which we will also associate the following initial condition:  (2.20)u(t0)=u0.
Note that since Bα-1 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.
LetI:=[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 Hs,if
(2.21)u(⋅,ω;t0,u0)∈C(I;Hs),P-a.s.ω∈Ω,
moreover, it satisfies in Hs the following integral equation:
(2.22)u(t,ω;t0,u0)=e-Aα(t-t0)v0-∫t0te-Aα(t-s)(Bα-1f(u)-βWA(s))ds+WA(t),P-a.s.ω∈Ω,
where WA(t) is called stochastic convolution (see Section 3 for details), β is a positive constant chosen in Section 3 and v0=u0-WA(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) Letα=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) Letα∈(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 B⊂H1.


## 3. Stochastic Convolution

LetWA(t) be the unique solution of linear equation (3.1)du+(Aα+β)udt=Bα-1A-1dW,
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, (3.2)WA(t)=∫-∞te-(t-s)(Aα+β)Bα-1A-1dW(s).Some regularity properties satisfied byWA(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)∈ℝ×G for any γ∈[0,δ/2).Proof.
We only consider the casen=3. For the sake of simplicity, we also assume that G=∏i=13(0,π). The eigenvectors of A can be given explicitly as follows:
(3.3)wk(x)=(2π)3/2cos⁡k1x1cos⁡k2x2cos⁡k3x3,x=(x1,x2,x3)∈R3,
with corresponding eigenvalues
(3.4)μk=k12+k22+k32=|k|2,k∈(Z+)3,
where k=(k1,k2,k3) varies in (ℤ+)3. Using (2.14), we find that
(3.5)WA(t,x)=∑k∈(Z+)3(αk∫-∞te-(t-s)(ηk+β)1αμk+1-αdβk(s))wk(x),
where ηk=μk2/(αμk+1-α), and hence,
(3.6)∇WA(t,x)-∇WA(t,y)=∑k∈(Z+)3(αk∫-∞te-(t-s)(ηk+β)(1/(αμk+1-α))dβk(s))(∇wk(x)-∇wk(y)),E(|∇WA(t,x)-∇WA(t,y)|2)≤∑k∈(Z+)3αk(αμk+1-α)2∫-∞te-2(t-s)(ηk+β)ds|∇wk(x)-∇wk(y)|2.
For anyγ∈[0,1], one trivially verifies that there is a constant cγ>0 independent of k such that for any k∈(ℤ+)3 and x,y∈G(3.7)|∇wk(x)-∇wk(y)|≤cγμk(1+γ)/2|x-y|γ.
Thus, we have
(3.8)E(|∇WA(t,x)-∇WA(t,y)|2)≤cγ22|x-y|2γ∑k∈(Z+)3αk(αμk+1-α)2ηk-1μk1+γ=cγ22|x-y|2γ∑k∈(Z+)3αk(αμk+1-α)2αμk+1-αμk2μk1+γ=cγ22|x-y|2γ∑k∈(Z+)3αkμkαμk+1-αμk-2+γ.
Now, lett,s∈ℝ. We may assume that t≥s. Then,
(3.9)E(|∇WA(t,x)-∇WA(s,x)|2)=∑k∈(Z+)3αk(αμk+1-α)2×(∫ste-2(ηk+β)(t-σ)dσ+∫-∞s[e-(ηk+β)(t-σ)-e-(ηk+β)(s-σ)]2dσ)|∇wk(x)|2=∑k∈(Z+)3αk(αμk+1-α)212(ηk+β)(1-e-2(ηk+β)(t-s))⋅|∇wk(x)|2.
Let 0≤γ≤1/2, and let
(3.10)cγ′=sup⁡r1,r2≥0|e-r1-e-r2||r1-r2|2γ.
Since the function g(r)=e-r is a Lipschitzoneon [0,∞), we always have cγ′<∞. Observe that
(3.11)E(|∇WA(t,x)-∇WA(s,x)|2)≤4γπ3cγ′|t-s|2γ∑k∈(Z+)3αk(αμk+1-α)2(ηk+β)2γ-1μk.≤4γπ3cγ′|t-s|2γ∑k∈(Z+)3αk(αμk+1-α)2ηk2γ-1μk=4γπ3cγ′|t-s|2γ∑k∈(Z+)3αk(μkαμk+1-α)2γ+1μk-2+2γ.
By (Q1), we know that Tr[Bα-1-δA-2+δQ]<∞ for some 0<δ≤1. Therefore, by (3.8) and (3.11), one deduces that there exists a constant cγ′′>0 such that
(3.12)E(|∇WA(t,x)-∇WA(s,y)|2)≤cγ′′(|x-y|2+|t-s|2)γ,∀(t,x),(s,y)∈R×G.
As WA(t,x)-WA(s,y) is a Gaussian process, we find that for each m∈ℤ+, there is a constant cγm>0 such that
(3.13)E(|∇WA(t,x)-∇WA(s,y)|2m)≤cγm(|x-y|2+|t-s|2)mγ.
Now, thanks to the well-known Kolmogorov test, one concludes that WA(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,(3.14)E(|WA(t)|22)≤M.Proof.
(3.15)E(|ΔWA(t)|2)=E(∑k∈(Z+)3αk∫-∞te-(ηk+β)(t-s)1αμk+1-αdβk(s)Δwk(x))2=∑k∈(Z+)3αk(αμk+1-α)2∫-∞te-2(ηk+β)(t-s)ds|Δwk(x)|2≤∑k∈(Z+)3αk(αμk+1-α)212(ηk+β)|Δwk(x)|2≤12(η1+β)∑k∈(Z+)3αk(μkαμk+1-α)2.
Since Tr[Bα-2Q]<∞, 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)∈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,
(3.16)E(|WA(t)|2+σ2)≤M.


## 4. Stochastic dissipativeness inH1

It is well known that in the deterministic case without forcing terms,(4.1)J(u)=12|∇u|2+∫GF(u)dx
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 thatu 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  (4.2)dJ(u)=(Ju(u),du)+12Tr⁡[Juu(u)Bα-2A-2Q]dt=(Ju(u),Bα-1A-1dW)-(Ju(u),Bα-1Au+Bα-1f(u))dt+12Tr⁡[Juu(u)Bα-2A-2Q]dt,
where Ju,Juu denote, respectively, the first and second derivative of J. Since (4.3)Ju(u)=Au+f(u),
there exists C1>0 such that for α=0,  (4.4)(Ju(u),Bα-1Au+Bα-1f(u))=|Au+f(u)|12≥λ12|Au+f(u)|-12=λ12(Au+f(u),u+A-1f(u))=λ12(|u|12+|f(u)|-12+2(f(u),u))≥dλ12(|u|12+∫GF(u)dx)-C1=dλ12J(u)-C1,
where d=min{1,4pa2p-1}. And for 0<α≤1,  (4.5)(Ju(u),Bα-1Au+Bα-1f(u))=(Au+f(u),Bα-1Au+Bα-1f(u))=|Au+f(u)|Bα-12≥λ12αλ1+1-α|Au+f(u)|-12≥dλ12αλ1+1-αJ(u)-C1,
where we have used (2.8). Simple computations show that (4.6)Juu(u)=A+f′(u),
and hence, (4.7)Tr⁡[Juu(u)Bα-2A-2Q]=Tr⁡[ABα-2A-2Q]+∑i=1∞(Di∫Gf′(u)wi2dx)=Tr⁡[Bα-2A-1Q]+∑i=1∞(Di∫Gf′(u)wi2dx),
where {wi}i=1∞ is the orthonormal basis of H as in (2.14), and Di=αi/(αλi+1-α)2.We infer from (3.3) that (4.8)|wi|L∞≤C2,
where C2>0 depends only on G. Therefore, (4.9)|∫Gf′(u)wi2dx|≤C22∫G|f′(u)|dx.
Set C3 such that (4.10)|f′(s)|≤2(2p-1)a2p-1s2p-2+C3,s∈R,
then(4.11)|∫Gf′(u)wi2dx|≤C22(2(2p-1)a2p-1∫Gu2p-2dx+C3|G|)≤14pa2p-1∫Gu2pdx+C4,
where C4 depends on f, p, and G. Let C5 satisfy (4.12)F(s)≥14pa2p-1s2p-C5|G|,s∈R,
then (4.13)|∫Gf′(u)wi2dx|≤J(u)+C4+C5.
Finally,  (4.14)Tr⁡[Juu(u)Bα-2A-2Q]≤Tr⁡[Bα-2A-1Q]+Tr⁡[Bα-2A-2Q](J(u)+C4+C5).
Since (4.15)E(Ju(u),Bα-1A-1dW)=0,
we have from (4.2) that (4.16)ddtE(J(u))=E(Ju(u),-Bα-1A(u)-Bα-1(u))+12E(Tr⁡[Juu(u)Bα-2A-2Q]).
Further, by (4.4), (4.5) and (4.14), it holds that(4.17)(ddt)E(J(u))≤-(D-12Tr⁡[Bα-2A-2Q])E(J(u))+Tr⁡[Bα-2A-1Q]+Tr⁡[Bα-2A-2Q](C4+C5)+C1,
where D=min{dλ12,dλ12/(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.
LetW be a H-valued Q-Wiener  process with
(4.18)Tr⁡[Bα-2A-1Q]<+∞,Tr⁡[Bα-2A-2Q]≤2D,
and let u(t) be the mild solution to (2.19)-(2.20). Then,
(4.19)E(J(u(t)))≤E(J(u0))+CQ,t∈[t0,∞),
where
(4.20)CQ=Tr⁡[Bα-2A-1Q]+Tr⁡[Bα-2A-2Q](C4+C5)+C1D-(1/2)Tr⁡[Bα-2A-2Q].As a consequence, we immediately obtain the following basic result.Corollary 4.2.
LetW be a H-valued Q-Wiener  process with
(4.21)Tr⁡[Bα-2A-1Q]<+∞,Tr⁡[Bα-2A-2Q]≤2D.
Then, there exists a continuous nonnegative functionΨ(r) such that for any solution u(t) of (2.19)-(2.20), one has
(4.22)E(|u(t)|12)≤Ψ(E(|u0|12)),∀t∈[t0,∞).


## 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: (5.1)v(t)=u(t)-WA(t).We observe thatv(t) satisfies the following system:  (5.2)dvdt+(Aα-β)v+Bα-1f(v+WA)=0,v(t0)=u0-WA(t0).
Let (5.3)G(v,t)=-Bα-1f(v+WA)+βWA,v0=u0-WA(t0).
Then, (5.2) reads  (5.4)dvdt+Aαv=G(v,t),v(t0)=v0.
To prove Theorem 2.3, it suffices to establish some corresponding existence results for the nonautonomous system (5.4).Definition 5.1.
LetI:=[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 Hs, if(5.5)v(⋅,ω;t0,v0)∈C(I;Hs),P-a.s.ω∈Ω,
and satisfies in Hs the following integral equation:
(5.6)v(t,ω;t0,v0)=e-Aα(t-t0)v0-∫t0te-Aα(t-s)(Bα-1f(u)-βWA(s))ds,P-a.s.ω∈Ω.Theorem 5.2.
Letα=0.  Suppose that the Hypothesis (Q2) is satisfied.
Then, for everyu0∈H2, there is a unique globally defined mild solution v(t,ω;t0,v0) of (5.4) in H2 with
(5.7)v(t,ω;t0,v0)∈C([t0,∞);H2)∩Cloc⁡0,1-r((t0,);H4r)∩C((t0,∞);H4),
for all 0≤r<1.Proof.
We only consider the case wheren=3. First, it is easy to verify that  P-a.s.
(5.8)G(v,t)∈CLip;γ(H2×[t0,∞),H).
Indeed, by Lemma 3.3, we see that WA(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
(5.9)|(G(v1,t1)-G(v2,t2))|≤C1(|v1-v2|2+|WA(t1)-WA(t2)|2)≤C2(ω)(|v1-v2|2+|t1-t2|γ),P-a.s.
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.
(5.10)v(t,ω;t0,v0)=e-A2(t-t0)v0-∫t0te-A2(t-s)(Af(u(s))-βWA(s))ds,v(t,ω;t0,v0)∈C([t0,T);H2)∩Cloc⁡0,1-r((t0,T);H4r)∩C((t0,T);H4),
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
(5.11)dvdt+A2v+Af(u)-βWA=0,v(t0)=v0.
In what follows, we show T=∞, thus proving the theorem.
Simple computations yields(5.12)|Δf(u)|≤|f′(u)|L∞|Δu|+|f′′(u)|L∞|∇u|L42.
Since f is a polynomial of degree 3, there exist κ1 and κ2 such that
(5.13)|f′(s)|≤κ1(1+|s|2),|f′′(s)|≤κ2(1+|s|),∀s∈R.
Therefore,
(5.14)|f′(u)|L∞|Δu|≤κ1(1+|u|L∞2)|Δu|≤2κ1(1+|v|L∞2+|WA|L∞2)(|Δv|+|ΔWA|).
By the Nirenberg-Gagiardo inequality, there exist C3,C4,C5>0 such that
(5.15)|u|L∞2≤C3|Δu|2,u∈H2,|u|L∞2≤C4|Δ2u|1/3|u|L65/3,u∈H4,|Δu|≤C5|Δ2u|1/2|∇u|1/2,u∈H4.
Hence,
(5.16)|f′(u)|L∞|Δu|≤2κ1(1+|v|L∞2+|WA|L∞2)(|Δv|+|ΔWA|)≤2κ1(1+C4|Δ2v|1/3|v|L65/3+C3|ΔWA|2)(C5|∇v|1/2|Δ2v|1/2+|ΔWA|).
By (Q2) and Lemma 3.2, we know that for  P-a.s. ω∈Ω, there exists an R1(ω)>0 such that |ΔWA(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
(5.17)|v|L65/3≤C6(ω),|∇v|1/2≤C7(ω),
where the continuous imbedding H1↪L6 is used. Consequently, we have
(5.18)|f′(u)|L∞|Δu|≤C8(ω)(1+|Δ2v|1/3+R1(ω))(|Δ2v|1/2+R1(ω)),P-a.s.ω∈Ω.
Similarly forP-a.s. ω∈Ω, one easily deduces that there exists C9(ω)>0 such that
(5.19)|f′′(u)|L∞|Δu|≤C9(ω)(1+|Δ2v|1/6+R1(ω))(|Δ2v|1/4+R1(ω)).
It then follows from (5.12) that for  P-a.s. ω∈Ω,
(5.20)|Δf(u)|≤C8(ω)(1+|Δ2v|1/3+R1(ω))(|Δ2v|1/2+R1(ω))+C9(ω)(1+L3|Δ2v|1/6+R1(ω))(|Δ2v|1/4+R1(ω))≤C10(ω)(1+|Δ2v|5/6).
Now, taking theL2 inner-product of equation (5.11) with Δ2v, one obtains
(5.21)12ddt|Δv|2+|Δ2v|2≤|∫GΔf(u)Δ2vdx|+β|∫GWAΔ2vdx|≤14|Δ2v|2+|Δf(u)|2+14|Δ2v|2+β2|WA|2≤12|Δ2v|2+|Δf(u)|2+β2λ1-2|ΔWA|2.
By (5.20), we deduce that  P-a.s.
(5.22)ddt|Δv|2+|Δ2v|2≤C11(ω)(1+|Δ2v|5/3).
Furthermore, by Young's inequality and |Δ2v|2≥λ12|Δv|2, we know that there exists C12(ω)>0 such that  P-a.s.
(5.23)ddt|Δv|2≤-λ122|Δv|2+C12(ω).
Applying the gronwall lemma on (5.23), one gets
(5.24)|Δv|2≤2C12(ω)λ12,P-a.s.ω∈Ω.
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.
Letα∈(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
(5.25)v(t,ω;t0,v0)∈C([t0,∞);H1)∩Cloc⁡0,1-r((t0,∞);H2r)∩C((t0,∞);H2),
for 0≤r<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 e-Aαt. It is easy to verify by Lemma 3.1 that  P-a.s.
(5.26)G(v,t)∈CLip;γ(H1×[t0,∞),H).
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
(5.27)v(t,ω;t0,v0)∈C([t0,T);H1)∩Cloc⁡0,1-r((t0,T);H2r)∩C((t0,T);H2),
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
(5.28)Bαdvdt+Av=-f(v+WA)+βBαWA,v(t0)=v0.
Multiplying (5.28) by Av, one gets
(5.29)12ddt|v|1,Bα2+|v|22+(f(v+WA),Av)+β(BαWA,Av)=0.
We observe that
(5.30)(f(v+WA),Av)=∫G∇f(v+WA)∇vdx=∫Gf′(v+WA)|∇v|2dx+∫Gf′(v+WA)∇WA∇vdx.
We take C1′ and C2′ such that
(5.31)f′(x)≥2p-12a2p-1x2p-2-C1′,|f′(x)|≤2(2p-1)a2p-1x2p-2+C′2,
for all x∈ℝ. Then,
(5.32)(f(v+WA),Av)≥2p-12a2p-1∫G|v+WA|2p-2|∇v|2dx-C1′∫G|∇v|2dx-2(2p-1)a2p-1∫G|v+WA|2p-2|∇v||∇WA|dx-C2′∫G|∇v||∇WA|dx≥14(2p-1)a2p-1∫G|v+WA|2p-2|∇v|2dx-2C1′∫G|∇v|2dx-C3′(∫G|∇WA|2pdx+∫G|∇WA|2dx),
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 α≤Cα≤M1 such that
(5.33)(BαWA,Av)≤Cα(∫G|∇WA|2dx+∫G|∇v|2dx).
Combining the last two inequalities together, we deduce that there exists constants C4′,C5′>0 such that
(5.34)12ddt|v|1,Bα2+|v|22+14(2p-1)a2p-1∫G|v+WA|2p-2|∇v|2dx≤2C4′∫G|∇v|2dx+C5′(∫G|∇WA|2pdx+∫G|∇WA|2dx).
In view of (2.7), there exists α≤Cα′≤M1 such that
(5.35)12Cα′ddt|v|12+|v|22+14(2p-1)a2p-1∫G|v+WA|2p-2|∇v|2dx≤2C4′|v|12+C5′(∫G|∇WA|2pdx+∫G|∇WA|2dx).
Using the gronwall lemma on (5.35), the following inequality holds:
(5.36)|v|12≤2e4C4′/Cα′C′4|v0|12+2e4C4′/Cα′∫t0tC5′(∫G|∇WA(s)|2pdx+∫G|∇WA(s)|2dx)ds.
Lemma 3.1 guarantees that  P-a.s.
(5.37)∫t0t∫G|∇WA(s)|2pdxds<+∞,∫t0t∫G|∇WA(s)|2dxds<+∞.
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 Theorem2.3 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 Theorem2.3, the mapping u0↦u(t,ω;t0,u0) defines a stochastic flow Sα(t,s;ω),(6.1)Sα(t,s;ω)u0=u(t,ω;s,u0),α∈[0,1].
Notice that  P-a.s.   (i)
Sα(t,s;ω)=Sα(t,r;ω)∘Sα(r,s;ω),foralls≤r≤t,(ii)
S0(t,s;ω) is continuous in H2, and Sα(t,s;ω) is continuous in H1 for 0<α≤1.


### 6.2. Compactness Properties of Stochastic FlowSα(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 Sα(t,s;ω),  0<α≤1,  is uniformly compact at time 0.  More precisely, for all B⊂H1 bounded and each t0<0,  Sα(0,t0;ω)B is  P-a.s.  relatively compact in H1.Proof.
(i) LetB⊂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 |WA(t)|2+σ≤R2(ω), t∈ℝ. Define B̂=B∪B2+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, B̂⊂H2 is P-a.s. bounded, and
(6.2)S0(0,t0;ω)B⊂{eA2t0v0-∫t00eA2sG(v(s),s)ds+WA(0),v0∈B̂}⊂N1+N2+N3+N4,P-a.s., where
(6.3)N1=eA2t0B̂,N2={∫-δ0eA2sG(v(s),s)ds,v0∈B̂},N3=e-A2δ{∫t0-δeA2(s+δ)G(v(s),s)ds,v0∈B̂},N4=B2+σ(0,R2(ω)),
and δ is an arbitrary constant satisfying 0<δ<-t0.Since fort>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.Forv0∈B̂, one has
(6.4)|∫-δ0eA2(s-t0)G(v(s),s)ds|2=|∫-δ0AeA2(s-t0)G(v(s),s)ds|.
Since A2 is a positive sectorial operator on H, there exists a constant MA>0 such that
(6.5)|Ae-A2t|L(H2)≤MAt-1/2,∀t≥0.
Recall that G(v,t)∈CLip;γ(H2×[t0,+),H). So there is a K0(ω)>0 such that P-a.s.
(6.6)|G(v,t)|≤K0(ω),∀(v,t)∈B̂×[-δ,0].
Therefore
(6.7)|∫-δ0eA2sG(v(s),s)ds|2≤K0(ω)MA∫-δ0(-s)1/2ds=12K0(ω)MAδ1/2.
It follows that
(6.8)κ(N2)≤diam⁡H2(N2)≤K0MAδ1/2,
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
(6.9)κ(S0(0,t0;ω)B̂)≤κ(N1)+κ(N2)+κ(N3)+κ(N4)≤κ(N2)≤K0MAδ1/2.
Letting δ→0, one immediately concludes that P-a.s. κ(S0(0,t0;ω)B̂)=0, hence S0(0,t0;ω)B̂ is relatively compact.(ii) The proof of the compactness result forSα(t,s;ω) (0<α≤1) is fully analogous, and is thus omitted.


### 6.3. The Random Attractors

Now, we show that the systemSα(t,s;ω) possesses a random attractor 𝒜α(ω) for every α∈[0,1].Proof of Theorem2.4.
We infer from the proofs of Theorem5.2 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
(6.10)B0={v:|Δv|2≤2C12(ω)λ12}∪B2+σ(0,R2(ω)),
and for Sα(t,s;ω) (0<α≤1), for any t0<0 we can define an absorbing set for Sα(t,t0;ω) at time 0 by
(6.11)Bα=B1(0,Ψ),
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


## 6.1. Stochastic Flows

Thanks to Theorem2.3, the mapping u0↦u(t,ω;t0,u0) defines a stochastic flow Sα(t,s;ω),(6.1)Sα(t,s;ω)u0=u(t,ω;s,u0),α∈[0,1].
Notice that  P-a.s.   (i)
Sα(t,s;ω)=Sα(t,r;ω)∘Sα(r,s;ω),foralls≤r≤t,(ii)
S0(t,s;ω) is continuous in H2, and Sα(t,s;ω) is continuous in H1 for 0<α≤1.


## 6.2. Compactness Properties of Stochastic FlowSα(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 Sα(t,s;ω),  0<α≤1,  is uniformly compact at time 0.  More precisely, for all B⊂H1 bounded and each t0<0,  Sα(0,t0;ω)B is  P-a.s.  relatively compact in H1.Proof.
(i) LetB⊂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 |WA(t)|2+σ≤R2(ω), t∈ℝ. Define B̂=B∪B2+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, B̂⊂H2 is P-a.s. bounded, and
(6.2)S0(0,t0;ω)B⊂{eA2t0v0-∫t00eA2sG(v(s),s)ds+WA(0),v0∈B̂}⊂N1+N2+N3+N4,P-a.s., where
(6.3)N1=eA2t0B̂,N2={∫-δ0eA2sG(v(s),s)ds,v0∈B̂},N3=e-A2δ{∫t0-δeA2(s+δ)G(v(s),s)ds,v0∈B̂},N4=B2+σ(0,R2(ω)),
and δ is an arbitrary constant satisfying 0<δ<-t0.Since fort>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.Forv0∈B̂, one has
(6.4)|∫-δ0eA2(s-t0)G(v(s),s)ds|2=|∫-δ0AeA2(s-t0)G(v(s),s)ds|.
Since A2 is a positive sectorial operator on H, there exists a constant MA>0 such that
(6.5)|Ae-A2t|L(H2)≤MAt-1/2,∀t≥0.
Recall that G(v,t)∈CLip;γ(H2×[t0,+),H). So there is a K0(ω)>0 such that P-a.s.
(6.6)|G(v,t)|≤K0(ω),∀(v,t)∈B̂×[-δ,0].
Therefore
(6.7)|∫-δ0eA2sG(v(s),s)ds|2≤K0(ω)MA∫-δ0(-s)1/2ds=12K0(ω)MAδ1/2.
It follows that
(6.8)κ(N2)≤diam⁡H2(N2)≤K0MAδ1/2,
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
(6.9)κ(S0(0,t0;ω)B̂)≤κ(N1)+κ(N2)+κ(N3)+κ(N4)≤κ(N2)≤K0MAδ1/2.
Letting δ→0, one immediately concludes that P-a.s. κ(S0(0,t0;ω)B̂)=0, hence S0(0,t0;ω)B̂ is relatively compact.(ii) The proof of the compactness result forSα(t,s;ω) (0<α≤1) is fully analogous, and is thus omitted.


## 6.3. The Random Attractors

Now, we show that the systemSα(t,s;ω) possesses a random attractor 𝒜α(ω) for every α∈[0,1].Proof of Theorem2.4.
We infer from the proofs of Theorem5.2 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
(6.10)B0={v:|Δv|2≤2C12(ω)λ12}∪B2+σ(0,R2(ω)),
and for Sα(t,s;ω) (0<α≤1), for any t0<0 we can define an absorbing set for Sα(t,t0;ω) at time 0 by
(6.11)Bα=B1(0,Ψ),
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



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*Source: 670786-2011-07-12.xml*