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((1βˆ’π›Ό)π‘’βˆ’π›ΌΞ”π‘’)+2ξ€Έξ€·π‘‘π‘’βˆ’Ξ”π‘“(𝑒)𝑑𝑑=π‘‘π‘Š,(π‘₯,𝑑)βˆˆπΊΓ—0ξ€Έ,∞,(1.1) subjected to homogeneous Dirichlet boundary conditions 𝑒𝑑(π‘₯,𝑑)=0,(π‘₯,𝑑)βˆˆπœ•πΊΓ—0ξ€Έ,∞,(1.2) in dimension 𝑛=1,2 or 3, where ∏𝐺=𝑛𝑖=1(0,𝐿𝑖) in 𝑅𝑛, and π›Όβˆˆ[0,1] is a parameter, 𝑓 is a polynomial of odd degree with a positive leading coefficient 𝑓(π‘₯)=2π‘βˆ’1ξ“π‘˜=1π‘Žπ‘˜π‘₯π‘˜,π‘Ž2π‘βˆ’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 π’œπ›Ό of the system has the same finite Hausdorff dimension for different parameter values 𝛼. One can also show that π’œπ›Ό 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 HΓΆlder 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 (β‹…,β‹…) and |β‹…| denote respectively the inner product and norm of 𝐻=𝐿2(𝐺). We define the linear operator 𝐴=βˆ’Ξ” with domain 𝐷(𝐴)=𝐻2⋂𝐻(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 |𝑒|𝑠=|𝐴𝑠/2𝑒| for real 𝑠. Note that 𝐻0=𝐿2(𝐺). It is well known that 𝐻𝑠 is a subspace of 𝐻𝑠(𝐺) and |β‹…|𝑠 is on 𝐻𝑠(𝐺) a norm equivalent to the usual one. Moreover, we have the following Poincare inequality and interpolation inequality: |𝑒|𝑠1β‰€πœ†βˆ’(𝑠2βˆ’π‘ 11)/2|𝑒|𝑠2,βˆ€π‘ 1,𝑠2βˆˆπ‘…,𝑠1<𝑠2,βˆ€π‘’βˆˆπ»π‘ 2,(2.1)|𝑒|πœŽπ‘ 1+(1βˆ’πœŽ)𝑠2≀|𝑒|πœŽπ‘ 1|𝑒|𝑠1βˆ’πœŽ2[],,𝜎∈0,1(2.2) where πœ†1 is the first eigenvalue of 𝐴.

We can define π΄βˆ’1βˆΆπ»β†’π·(𝐴) to be the Green’s operator for 𝐴. Thus, 𝑣=π΄βˆ’1π‘€βŸΊπ΄π‘£=𝑀.(2.3) By Rellich’s Theorem, we know that π΄βˆ’1 is compact, and 𝐴∢𝐷(𝐴)→𝐻 is a linear and bounded operator. Finally, we introduce the invertible operator π΅π›ΌβˆΆπ»π‘ β†’π»π‘ , π‘ βˆˆβ„ defined by π΅π›ΌβˆΆ=𝛼𝐼+(1βˆ’π›Ό)π΄βˆ’1.(2.4) For each π›Όβˆˆ(0,1] and 𝛽β‰₯0, we know that π΅π›½π›ΌβˆΆπ»π‘ β†’π»π‘  is bounded and has a bounded inverse (see [10, 11]). We also define the operator π΄π›ΌβˆΆ=π΅π›Όβˆ’1𝐴 with domain𝐷𝐴𝛼=𝐷𝐴𝐷(𝐴)if𝛼>0,0ξ€Έ=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|𝐡𝛼≀𝑀11/2𝛼|𝑣|,π‘£βˆˆπ»,(2.6)1/2|𝑣|1≀|𝑣|1,𝐡𝛼≀𝑀21/2|𝑣|1,π‘£βˆˆπ»1ξ‚΅πœ†,(2.7)1π›Όπœ†1ξ‚Ά+1βˆ’π›Ό1/2|𝑣|≀|𝑣|π΅π›Όβˆ’1≀𝑀31/2|𝑣|,π‘£βˆˆπ»,(2.8) where |𝑣|π΅π›Όξ€·βˆΆ=𝑣,𝐡𝛼𝑣1/2,|𝑣|1,π΅π›Όξ€·π΄βˆΆ=1/2𝑣,𝐡𝛼𝐴1/2𝑣1/2,|𝑣|π΅π›Όβˆ’1ξ€·βˆΆ=𝑣,π΅π›Όβˆ’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 π΅π›Όβˆ’1/2 is bounded, there exists 𝑀3β‰₯0, such that |π΅π›Όβˆ’1/2|2≀𝑀3. Then, for any π‘£βˆˆπ», we have 𝑣,π΅π›Όβˆ’1𝑣=ξ€·π΅π›Όβˆ’1/2𝑣,π΅π›Όβˆ’1/2𝑣=||π΅π›Όβˆ’1/2𝑒||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<πœ†1β‰€πœ†2β‰€β‹―β‰€πœ†π‘˜β‰€β‹―(2.11) denote the eigenvalues of 𝐴, repeated with the respective multiplicity, and the corresponding unit eigenvector is denoted by {π‘€π‘˜}βˆžπ‘˜=1, which forms an orthonormal basis for 𝐻. We have ξ€·π‘€π‘˜,π΅π›Όβˆ’1π‘€π‘˜ξ€Έ=πœ†π‘˜π›Όπœ†π‘˜β‰₯πœ†+1βˆ’π›Ό1π›Όπœ†1ξ€·β„€+1βˆ’π›Ό,π‘˜βˆˆ+ξ€Έ.(2.12) Since π‘£βˆˆπ», there exist {π‘π‘˜}βˆžπ‘˜=1βŠ‚β„, such that βˆ‘π‘£=+βˆžπ‘˜=1π‘π‘˜π‘€π‘˜. Consequently, 𝑣,π΅π›Όβˆ’1𝑣=+βˆžξ“π‘˜=1π‘π‘˜π‘€π‘˜,π΅π›Όβˆ’1+βˆžξ“π‘˜=1π‘π‘˜π‘€π‘˜ξƒͺ=+βˆžξ“π‘˜=1ξ€·π‘π‘˜π‘€π‘˜,π΅π›Όβˆ’1π‘π‘˜π‘€π‘˜ξ€Έ=+βˆžξ“π‘˜=1πœ†π‘˜π›Όπœ†π‘˜π‘+1βˆ’π›Ό2π‘˜β‰₯πœ†1π›Όπœ†1+1βˆ’π›Ό+βˆžξ“π‘˜=1𝑏2π‘˜=πœ†1π›Όπœ†1+1βˆ’π›Ό|𝑣|2,(2.13) which finishes the proof.

2.2. Assumptions on the Noise

The stochastic process π‘Š(𝑑), defined on a probability space (Ξ©,β„±,𝐏), is a two-side in time Wiener process on 𝐻 which is given by the expansions π‘Š(𝑑)=βˆžξ“π‘˜=0βˆšπ›Όπ‘˜π›½π‘˜(𝑑)π‘€π‘˜,(2.14) where {π‘€π‘˜}βˆžπ‘˜=1 is a basis of 𝐻 consisting of unit eigenvectors of 𝐴, {π›Όπ‘˜}βˆžπ‘˜=1 is a bounded sequence of nonnegative numbers, and π›½π‘˜1(𝑑)=βˆšπ›Όπ‘˜ξ€·π‘Š(𝑑),π‘€π‘˜ξ€Έ,π‘˜βˆˆβ„•(2.15) is a sequence of mutually independent real valued standard Brownian motions in a fixed probability space (Ξ©,β„±,𝐏) adapted to a filtration {ℱ𝑑}𝑑β‰₯0.

For convenience, we will define the covariance operator 𝑄 on 𝐻 as follows: π‘„π‘€π‘˜=π›Όπ‘˜π‘€π‘˜,π‘˜βˆˆβ„•.(2.16) The process π‘Š(𝑑) will be called as the 𝑄-Wiener process. We need to impose on 𝑄 one of the following assumptions: (Q1)Tr[π΅π›Όβˆ’1βˆ’π›Ώπ΄βˆ’2+𝛿𝑄]<∞(forsome0<𝛿≀1), (Q1*)Tr[π΅π›Όβˆ’2π΄βˆ’1𝑄]<∞,and Tr[π΅π›Όβˆ’2π΄βˆ’2𝑄]≀2𝐷,(Q2)Tr[π΅π›Όβˆ’1βˆ’π›Ώπ΄βˆ’1+𝛿𝑄]<∞(for some0<𝛿≀1),Tr[π΅π›Όβˆ’2𝑄]<∞, and Tr[π΅π›Όβˆ’2π΄βˆ’2𝑄]≀2𝐷,(Q2*)Tr[π΅π›Όβˆ’1βˆ’π›Ώπ΄βˆ’1+𝛿𝑄]<∞,Tr[π΅π›Όβˆ’2π΄πœŽπ‘„]<∞ (for some 0<𝛿≀1 and 𝜎>0), and Tr[π΅π›Όβˆ’2π΄βˆ’2𝑄]≀2𝐷,

where 𝐷 is given in Section 4. It is obvious that ξ€·ππŸβˆ—ξ€ΈβŸΉξ€·(𝐐𝟐),ππŸβˆ—ξ€ΈβŸΉ(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 𝐴𝑑𝑒+𝛼𝑒+π΅π›Όβˆ’1𝑓(𝑒)𝑑𝑑=π΅π›Όβˆ’1π΄βˆ’1π‘‘π‘Š,(2.19) with which we will also associate the following initial condition: 𝑒𝑑0ξ€Έ=𝑒0.(2.20) Note that since π΅π›Όβˆ’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. Let 𝐼∢=[𝑑0,𝑑0+𝜏) be an interval in ℝ. We say that a stochastic process 𝑒(𝑑,πœ”;𝑑0,𝑒0) is a mild solution of the system (2.19)-(2.20) in 𝐻𝑠, if 𝑒⋅,πœ”;𝑑0,𝑒0ξ€Έξ€·βˆˆπΆπΌ;𝐻𝑠,𝐏-a.s.πœ”βˆˆΞ©,(2.21) moreover, it satisfies in 𝐻𝑠 the following integral equation: 𝑒𝑑,πœ”;𝑑0,𝑒0ξ€Έ=π‘’βˆ’π΄π›Ό(π‘‘βˆ’π‘‘0)𝑣0βˆ’ξ€œπ‘‘π‘‘0π‘’βˆ’π΄π›Ό(π‘‘βˆ’π‘ )ξ€·π΅π›Όβˆ’1𝑓(𝑒)βˆ’π›½π‘Šπ΄(𝑠)𝑑𝑠+π‘Šπ΄(𝑑),𝐏-a.s.πœ”βˆˆΞ©,(2.22) where π‘Šπ΄(𝑑) is called stochastic convolution (see Section 3 for details), 𝛽 is a positive constant chosen in Section 3 and 𝑣0=𝑒0βˆ’π‘Šπ΄(𝑑0).

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 𝑒0∈𝐻2, there is a unique maximally defined mild solution 𝑒(𝑑,πœ”;𝑑0,𝑒0) of (2.19)-(2.20) in 𝐻2 for all π‘‘βˆˆ[𝑑0,∞).
(𝑖𝑖) Let π›Όβˆˆ(0,1], and, the hypothesis (𝐐𝟏) be satisfied. Then for every 𝑒0∈𝐻1, there is a unique maximally defined mild solution 𝑒(𝑑,πœ”;𝑑0,𝑒0) of (2.19)-(2.20) in 𝐻1 for all π‘‘βˆˆ[𝑑0,∞).

Theorem 2.4. (i) Let 𝛼=0, and, the hypothesis (ππŸβˆ—) be satisfied. Then the stochastic flow associated with (2.19)-(2.20) has a compact stochastic attractor π’œ0(πœ”)βŠ‚π»2 at time 0, which pullback attracts every bounded deterministic set π΅βŠ‚π»2.
(ii) Let π›Όβˆˆ(0,1], and, the hypothesis (ππŸβˆ—) be satisfied. Then the stochastic flow associated with (2.19)-(2.20) has a compact stochastic attractor π’œπ›Ό(πœ”)βŠ‚π»1 at time 0, which pullback attracts every bounded deterministic set π΅βŠ‚π»1.

3. Stochastic Convolution

Let π‘Šπ΄(𝑑) be the unique solution of linear equation 𝐴𝑑𝑒+𝛼+𝛽𝑒𝑑𝑑=π΅π›Όβˆ’1π΄βˆ’1π‘‘π‘Š,(3.1) where 𝛽 is a positive constant to be further determined. Then, π‘Šπ΄(𝑑) is an ergodic and stationary process [9, 12] called the stochastic convolution. Moreover, π‘Šπ΄(ξ€œπ‘‘)=π‘‘βˆ’βˆžπ‘’βˆ’(π‘‘βˆ’π‘ )(𝐴𝛼+𝛽)π΅π›Όβˆ’1π΄βˆ’1π‘‘π‘Š(𝑠).(3.2)

Some regularity properties satisfied by π‘Šπ΄(𝑑) are given below.

Lemma 3.1. Assume that (𝐐𝟏) holds. Then, βˆ‡π‘Šπ΄(𝑑) has a version which is 𝛾-HΓΆlder continuous with respect to (𝑑,π‘₯)βˆˆβ„Γ—πΊ for any π›Ύβˆˆ[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œ‹). The eigenvectors of 𝐴 can be given explicitly as follows: π‘€π‘˜ξ‚€2(π‘₯)=πœ‹ξ‚3/2cosπ‘˜1π‘₯1cosπ‘˜2π‘₯2cosπ‘˜3π‘₯3ξ€·π‘₯,π‘₯=1,π‘₯2,π‘₯3ξ€Έβˆˆβ„3,(3.3) with corresponding eigenvalues πœ‡π‘˜=π‘˜21+π‘˜22+π‘˜23=||π‘˜||2ξ€·β„€,π‘˜βˆˆ+ξ€Έ3,(3.4) where π‘˜=(π‘˜1,π‘˜2,π‘˜3) varies in (β„€+)3. Using (2.14), we find that π‘Šπ΄(𝑑,π‘₯)=π‘˜βˆˆ(β„€+)3ξ‚΅βˆšπ›Όπ‘˜ξ€œπ‘‘βˆ’βˆžπ‘’βˆ’(π‘‘βˆ’π‘ )(πœ‚π‘˜+𝛽)1π›Όπœ‡π‘˜+1βˆ’π›Όπ‘‘π›½π‘˜(𝑀𝑠)π‘˜(π‘₯),(3.5) where πœ‚π‘˜=πœ‡2π‘˜/(π›Όπœ‡π‘˜+1βˆ’π›Ό), and hence, βˆ‡π‘Šπ΄(𝑑,π‘₯)βˆ’βˆ‡π‘Šπ΄=(𝑑,𝑦)π‘˜βˆˆ(β„€+)3ξ‚΅βˆšπ›Όπ‘˜ξ€œπ‘‘βˆ’βˆžπ‘’βˆ’(π‘‘βˆ’π‘ )(πœ‚π‘˜+𝛽)(1/(π›Όπœ‡π‘˜+1βˆ’π›Ό))π‘‘π›½π‘˜(𝑠)βˆ‡π‘€π‘˜(π‘₯)βˆ’βˆ‡π‘€π‘˜(ξ€Έ,𝐄||𝑦)βˆ‡π‘Šπ΄(𝑑,π‘₯)βˆ’βˆ‡π‘Šπ΄||(𝑑,𝑦)2ξ‚β‰€ξ“ξ€·β„€π‘˜βˆˆ+ξ€Έ3π›Όπ‘˜ξ€·π›Όπœ‡π‘˜ξ€Έ+1βˆ’π›Ό2ξ€œπ‘‘βˆ’βˆžπ‘’βˆ’2(π‘‘βˆ’π‘ )(πœ‚π‘˜+𝛽)||π‘‘π‘ βˆ‡π‘€π‘˜(π‘₯)βˆ’βˆ‡π‘€π‘˜||(𝑦)2.(3.6)
For any π›Ύβˆˆ[0,1], one trivially verifies that there is a constant 𝑐𝛾>0 independent of π‘˜ such that for any π‘˜βˆˆ(β„€+)3 and π‘₯,π‘¦βˆˆπΊ||βˆ‡π‘€π‘˜(π‘₯)βˆ’βˆ‡π‘€π‘˜||(𝑦)β‰€π‘π›Ύπœ‡π‘˜(1+𝛾)/2||||π‘₯βˆ’π‘¦π›Ύ.(3.7) Thus, we have 𝐄||βˆ‡π‘Šπ΄(𝑑,π‘₯)βˆ’βˆ‡π‘Šπ΄||(𝑑,𝑦)2≀𝑐2𝛾2||||π‘₯βˆ’π‘¦2π›Ύξ“ξ€·β„€π‘˜βˆˆ+ξ€Έ3π›Όπ‘˜ξ€·π›Όπœ‡π‘˜ξ€Έ+1βˆ’π›Ό2πœ‚π‘˜βˆ’1πœ‡π‘˜1+𝛾=𝑐2𝛾2||||π‘₯βˆ’π‘¦2π›Ύξ“ξ€·β„€π‘˜βˆˆ+ξ€Έ3π›Όπ‘˜ξ€·π›Όπœ‡π‘˜ξ€Έ+1βˆ’π›Ό2π›Όπœ‡π‘˜+1βˆ’π›Όπœ‡2π‘˜πœ‡π‘˜1+𝛾=𝑐2𝛾2||||π‘₯βˆ’π‘¦2π›Ύξ“ξ€·β„€π‘˜βˆˆ+ξ€Έ3π›Όπ‘˜πœ‡π‘˜π›Όπœ‡π‘˜πœ‡+1βˆ’π›Όπ‘˜βˆ’2+𝛾.(3.8)
Now, let 𝑑,π‘ βˆˆβ„. We may assume that 𝑑β‰₯𝑠. Then, 𝐄||βˆ‡π‘Šπ΄(𝑑,π‘₯)βˆ’βˆ‡π‘Šπ΄||(𝑠,π‘₯)2=ξ“ξ€·β„€π‘˜βˆˆ+ξ€Έ3π›Όπ‘˜ξ€·π›Όπœ‡π‘˜ξ€Έ+1βˆ’π›Ό2Γ—ξ‚΅ξ€œπ‘‘π‘ π‘’βˆ’2(πœ‚π‘˜+𝛽)(π‘‘βˆ’πœŽ)ξ€œπ‘‘πœŽ+π‘ βˆ’βˆžξ€Ίπ‘’βˆ’(πœ‚π‘˜+𝛽)(π‘‘βˆ’πœŽ)βˆ’π‘’βˆ’(πœ‚π‘˜+𝛽)(π‘ βˆ’πœŽ)ξ€»2ξ‚Ά||π‘‘πœŽβˆ‡π‘€π‘˜||(π‘₯)2=ξ“ξ€·β„€π‘˜βˆˆ+ξ€Έ3π›Όπ‘˜ξ€·π›Όπœ‡π‘˜ξ€Έ+1βˆ’π›Ό212ξ€·πœ‚π‘˜ξ€Έξ€·+𝛽1βˆ’π‘’βˆ’2(πœ‚π‘˜+𝛽)(π‘‘βˆ’π‘ )ξ€Έβ‹…||βˆ‡π‘€π‘˜||(π‘₯)2.(3.9) Let 0≀𝛾≀1/2, and let π‘ξ…žπ›Ύ=supπ‘Ÿ1,π‘Ÿ2β‰₯0||π‘’βˆ’π‘Ÿ1βˆ’π‘’βˆ’π‘Ÿ2||||π‘Ÿ1βˆ’π‘Ÿ2||2𝛾.(3.10) Since the function g(π‘Ÿ)=π‘’βˆ’π‘Ÿ is a Lipschitzoneon [0,∞), we always have π‘ξ…žπ›Ύ<∞. Observe that 𝐄||βˆ‡π‘Šπ΄(𝑑,π‘₯)βˆ’βˆ‡π‘Šπ΄||(𝑠,π‘₯)2≀4π›Ύπœ‹3π‘ξ…žπ›Ύ|π‘‘βˆ’π‘ |2π›Ύξ“π‘˜βˆˆ(β„€+)3π›Όπ‘˜ξ€·π›Όπœ‡π‘˜ξ€Έ+1βˆ’π›Ό2ξ€·πœ‚π‘˜ξ€Έ+𝛽2π›Ύβˆ’1πœ‡π‘˜.≀4π›Ύπœ‹3π‘ξ…žπ›Ύ|π‘‘βˆ’π‘ |2π›Ύξ“π‘˜βˆˆ(β„€+)3π›Όπ‘˜ξ€·π›Όπœ‡π‘˜ξ€Έ+1βˆ’π›Ό2πœ‚π‘˜2π›Ύβˆ’1πœ‡π‘˜=4π›Ύπœ‹3π‘ξ…žπ›Ύ|π‘‘βˆ’π‘ |2π›Ύξ“π‘˜βˆˆ(β„€+)3π›Όπ‘˜ξ‚΅πœ‡π‘˜π›Όπœ‡π‘˜ξ‚Ά+1βˆ’π›Ό2𝛾+1πœ‡π‘˜βˆ’2+2𝛾.(3.11) By (𝐐𝟏), we know that Tr[π΅π›Όβˆ’1βˆ’π›Ώπ΄βˆ’2+𝛿𝑄]<∞ for some 0<𝛿≀1. Therefore, by (3.8) and (3.11), one deduces that there exists a constant π‘π›Ύξ…žξ…ž>0 such that 𝐄||βˆ‡π‘Šπ΄(𝑑,π‘₯)βˆ’βˆ‡π‘Šπ΄||(𝑠,𝑦)2ξ‚β‰€π‘π›Ύξ…žξ…žξ‚€||||π‘₯βˆ’π‘¦2+|π‘‘βˆ’π‘ |2𝛾,βˆ€(𝑑,π‘₯),(𝑠,𝑦)βˆˆβ„Γ—πΊ.(3.12) As π‘Šπ΄(𝑑,π‘₯)βˆ’π‘Šπ΄(𝑠,𝑦) is a Gaussian process, we find that for each π‘šβˆˆβ„€+, there is a constant π‘π‘šπ›Ύ>0 such that 𝐄||βˆ‡π‘Šπ΄(𝑑,π‘₯)βˆ’βˆ‡π‘Šπ΄||(𝑠,𝑦)2π‘šξ‚β‰€π‘π‘šπ›Ύξ‚€||||π‘₯βˆ’π‘¦2+|π‘‘βˆ’π‘ |2ξ‚π‘šπ›Ύ.(3.13) Now, thanks to the well-known Kolmogorov test, one concludes that π‘Šπ΄(𝑑,π‘₯) is (π›Ύβˆ’2/π‘š)-HΓΆlder continuous in (𝑑,π‘₯). Because π›Ύβˆˆ[0,1/2] and π‘šβˆˆβ„€+ 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 𝛽β‰₯𝛽0, 𝐄||π‘Šπ΄||(𝑑)22≀𝑀.(3.14)

Proof. 𝐄||Ξ”π‘Šπ΄(||𝑑)2ξ‚βŽ›βŽœβŽœβŽξ“=π„π‘˜βˆˆ(β„€+)3βˆšπ›Όπ‘˜ξ€œπ‘‘βˆ’βˆžπ‘’βˆ’(πœ‚π‘˜+𝛽)(π‘‘βˆ’π‘ )1π›Όπœ‡π‘˜+1βˆ’π›Όπ‘‘π›½π‘˜(𝑠)Ξ”π‘€π‘˜(⎞⎟⎟⎠π‘₯)2=ξ“π‘˜βˆˆ(β„€+)3π›Όπ‘˜ξ€·π›Όπœ‡π‘˜ξ€Έ+1βˆ’π›Ό2ξ€œπ‘‘βˆ’βˆžπ‘’βˆ’2(πœ‚π‘˜+𝛽)(π‘‘βˆ’π‘ )||π‘‘π‘ Ξ”π‘€π‘˜||(π‘₯)2β‰€ξ“ξ€·β„€π‘˜βˆˆ+ξ€Έ3π›Όπ‘˜ξ€·π›Όπœ‡π‘˜ξ€Έ+1βˆ’π›Ό212ξ€·πœ‚π‘˜ξ€Έ||+π›½Ξ”π‘€π‘˜||(π‘₯)2≀12ξ€·πœ‚1+π›½π‘˜βˆˆ(β„€+)3π›Όπ‘˜ξ‚΅πœ‡π‘˜π›Όπœ‡π‘˜ξ‚Ά+1βˆ’π›Ό2.(3.15) Since Tr[π΅π›Όβˆ’2𝑄]<∞, one can now easily choose a 𝛽 large enough so that 𝐄(|Ξ”π‘Šπ΄(𝑑)|2)≀𝑀, and the proof is complete.

Similarly, we can verify the following basic fact.

Lemma 3.3. Assume (𝐐𝟐) holds. Then, Ξ”π‘Šπ΄ has a version which is 𝛾-HΓΆlder continuous with respect to (𝑑,π‘₯)βˆˆβ„Γ—πΊ for any π›Ύβˆˆ[0,𝛿/2).

Lemma 3.4. Assume that (ππŸβˆ—) holds. Then, for any 𝑀>0, there exists 𝛽0 such that for all 𝛽β‰₯𝛽0, 𝐄||π‘Šπ΄||(𝑑)22+πœŽξ‚β‰€π‘€.(3.16)

4. Stochastic dissipativeness in 𝐻1

It is well known that in the deterministic case without forcing terms, 1𝐽(𝑒)=2||||βˆ‡π‘’2+ξ€œπΊπΉ(𝑒)𝑑π‘₯(4.1) is a Lyapunov functional of the system (i.e. (𝑑/𝑑𝑑)𝐽(𝑒)≀0), where 𝐹(𝑒) is the primitive function of 𝑓(𝑒) 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 𝑒 satisfies (2.19)-(2.20). As usual, we may assume in advance that 𝑒 is sufficiently regular so that all the computations can be performed rigorously. Applying the ItΓ΄ formula to 𝐽(𝑒), we obtain 𝐽𝑑𝐽(𝑒)=𝑒+1(𝑒),𝑑𝑒2𝐽Tr𝑒𝑒(𝑒)π΅π›Όβˆ’2π΄βˆ’2𝑄=𝐽𝑑𝑑𝑒(𝑒),π΅π›Όβˆ’1π΄βˆ’1ξ€Έβˆ’ξ€·π½π‘‘π‘Šπ‘’(𝑒),π΅π›Όβˆ’1𝐴𝑒+π΅π›Όβˆ’1ξ€Έ1𝑓(𝑒)𝑑𝑑+2𝐽Tr𝑒𝑒(𝑒)π΅π›Όβˆ’2π΄βˆ’2𝑄𝑑𝑑,(4.2) where 𝐽𝑒,𝐽𝑒𝑒 denote, respectively, the first and second derivative of 𝐽. Since 𝐽𝑒(𝑒)=𝐴𝑒+𝑓(𝑒),(4.3) there exists 𝐢1>0 such that for 𝛼=0, 𝐽𝑒(𝑒),π΅π›Όβˆ’1𝐴𝑒+π΅π›Όβˆ’1ξ€Έ=||||𝑓(𝑒)𝐴𝑒+𝑓(𝑒)21β‰₯πœ†21||||𝐴𝑒+𝑓(𝑒)2βˆ’1=πœ†21𝐴𝑒+𝑓(𝑒),𝑒+π΄βˆ’1𝑓(𝑒)=πœ†21ξ‚€|𝑒|21+||||𝑓(𝑒)2βˆ’1+2(𝑓(𝑒),𝑒)β‰₯π‘‘πœ†21ξ‚΅|𝑒|21+ξ€œπΊξ‚ΆπΉ(𝑒)𝑑π‘₯βˆ’πΆ1=π‘‘πœ†21𝐽(𝑒)βˆ’πΆ1,(4.4) where 𝑑=min{1,4π‘π‘Ž2π‘βˆ’1}. And for 0<𝛼≀1, 𝐽𝑒(𝑒),π΅π›Όβˆ’1𝐴𝑒+π΅π›Όβˆ’1ξ€Έ=𝑓(𝑒)𝐴𝑒+𝑓(𝑒),π΅π›Όβˆ’1𝐴𝑒+π΅π›Όβˆ’1ξ€Έ=||||𝑓(𝑒)𝐴𝑒+𝑓(𝑒)2π΅π›Όβˆ’1β‰₯πœ†21π›Όπœ†1||||+1βˆ’π›Όπ΄π‘’+𝑓(𝑒)2βˆ’1β‰₯π‘‘πœ†21π›Όπœ†1+1βˆ’π›Όπ½(𝑒)βˆ’πΆ1,(4.5) where we have used (2.8). Simple computations show that 𝐽𝑒𝑒(𝑒)=𝐴+π‘“ξ…ž(𝑒),(4.6) and hence, 𝐽Tr𝑒𝑒(𝑒)π΅π›Όβˆ’2π΄βˆ’2𝑄=Trπ΄π΅π›Όβˆ’2π΄βˆ’2𝑄+βˆžξ“π‘–=1ξ‚΅π·π‘–ξ€œπΊπ‘“ξ…ž(𝑒)𝑀2𝑖𝐡𝑑π‘₯=Trπ›Όβˆ’2π΄βˆ’1𝑄+βˆžξ“π‘–=1ξ‚΅π·π‘–ξ€œπΊπ‘“ξ…ž(𝑒)𝑀2𝑖,𝑑π‘₯(4.7) where {𝑀𝑖}βˆžπ‘–=1 is the orthonormal basis of 𝐻 as in (2.14), and 𝐷𝑖=𝛼𝑖/(π›Όπœ†π‘–+1βˆ’π›Ό)2.

We infer from (3.3) that ||𝑀𝑖||πΏβˆžβ‰€πΆ2,(4.8) where 𝐢2>0 depends only on 𝐺. Therefore, ||||ξ€œπΊπ‘“ξ…ž(𝑒)𝑀2𝑖||||𝑑π‘₯≀𝐢22ξ€œπΊ||π‘“ξ…ž||(𝑒)𝑑π‘₯.(4.9) Set 𝐢3 such that ||π‘“ξ…ž||(𝑠)≀2(2π‘βˆ’1)π‘Ž2π‘βˆ’1𝑠2π‘βˆ’2+𝐢3,π‘ βˆˆβ„,(4.10) then||||ξ€œπΊπ‘“ξ…ž(𝑒)𝑀2𝑖||||𝑑π‘₯≀𝐢22ξ‚΅2(2π‘βˆ’1)π‘Ž2π‘βˆ’1ξ€œπΊπ‘’2π‘βˆ’2𝑑π‘₯+𝐢3||𝐺||≀1π‘Ž4𝑝2π‘βˆ’1ξ€œπΊπ‘’2𝑝𝑑π‘₯+𝐢4,(4.11) where 𝐢4 depends on 𝑓, 𝑝, and 𝐺. Let 𝐢5 satisfy 1𝐹(𝑠)β‰₯π‘Ž4𝑝2π‘βˆ’1𝑠2π‘βˆ’πΆ5||𝐺||,π‘ βˆˆβ„,(4.12) then ||||ξ€œπΊπ‘“ξ…ž(𝑒)𝑀2𝑖||||𝑑π‘₯≀𝐽(𝑒)+𝐢4+𝐢5.(4.13) Finally, 𝐽Tr𝑒𝑒(𝑒)π΅π›Όβˆ’2π΄βˆ’2𝑄𝐡≀Trπ›Όβˆ’2π΄βˆ’1𝑄𝐡+Trπ›Όβˆ’2π΄βˆ’2𝑄𝐽(𝑒)+𝐢4+𝐢5ξ€Έ.(4.14) Since 𝐄𝐽𝑒(𝑒),π΅π›Όβˆ’1π΄βˆ’1ξ€Έπ‘‘π‘Š=0,(4.15) we have from (4.2) that 𝑑𝐽𝑑𝑑𝐄(𝐽(𝑒))=𝐄𝑒(𝑒),βˆ’π΅π›Όβˆ’1𝐴(𝑒)βˆ’π΅π›Όβˆ’1ξ€Έ+1(𝑒)2𝐄𝐽Tr𝑒𝑒(𝑒)π΅π›Όβˆ’2π΄βˆ’2𝑄.ξ€»ξ€Έ(4.16) Further, by (4.4), (4.5) and (4.14), it holds that𝑑1𝑑𝑑𝐄(𝐽(𝑒))β‰€βˆ’π·βˆ’2𝐡Trπ›Όβˆ’2π΄βˆ’2𝑄𝐡𝐄(𝐽(𝑒))+Trπ›Όβˆ’2π΄βˆ’1𝑄𝐡+Trπ›Όβˆ’2π΄βˆ’2𝑄𝐢4+𝐢5ξ€Έ+𝐢1,(4.17) where 𝐷=min{π‘‘πœ†21,π‘‘πœ†21/(π›Όπœ†1+1βˆ’π›Ό)}. This is precisely what we promised.

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

Lemma 4.1. Let π‘Š be a H-valued Q-Wiener process with 𝐡Trπ›Όβˆ’2π΄βˆ’1𝑄𝐡<+∞,Trπ›Όβˆ’2π΄βˆ’2𝑄≀2𝐷,(4.18) and let 𝑒(𝑑) be the mild solution to (2.19)-(2.20). Then, 𝐄𝐽𝑒(𝐽(𝑒(𝑑)))≀𝐄0ξ€Έξ€Έ+𝐢𝑄𝑑,π‘‘βˆˆ0ξ€Έ,∞,(4.19) where 𝐢𝑄=𝐡Trπ›Όβˆ’2π΄βˆ’1𝑄𝐡+Trπ›Όβˆ’2π΄βˆ’2𝑄𝐢4+𝐢5ξ€Έ+𝐢1ξ€Ίπ΅π·βˆ’(1/2)Trπ›Όβˆ’2π΄βˆ’2𝑄.(4.20)

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

Corollary 4.2. Let π‘Š be a H-valued Q-Wiener process with 𝐡Trπ›Όβˆ’2π΄βˆ’1𝑄𝐡<+∞,Trπ›Όβˆ’2π΄βˆ’2𝑄≀2𝐷.(4.21)
Then, there exists a continuous nonnegative function Ξ¨(π‘Ÿ) such that for any solution 𝑒(𝑑) of (2.19)-(2.20), one has 𝐄||||𝑒(𝑑)21𝐄||𝑒≀Ψ0||21𝑑,βˆ€π‘‘βˆˆ0ξ€Έ,∞.(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(𝑑)=𝑒(𝑑)βˆ’π‘Šπ΄(𝑑).(5.1)

We observe that 𝑣(𝑑) satisfies the following system: 𝑑𝑣+ξ€·π΄π‘‘π‘‘π›Όξ€Έβˆ’π›½π‘£+π΅π›Όβˆ’1𝑓𝑣+π‘Šπ΄ξ€Έπ‘£ξ€·π‘‘=0,0ξ€Έ=𝑒0βˆ’π‘Šπ΄ξ€·π‘‘0ξ€Έ.(5.2) Let 𝐺(𝑣,𝑑)=βˆ’π΅π›Όβˆ’1𝑓𝑣+π‘Šπ΄ξ€Έ+π›½π‘Šπ΄,𝑣0=𝑒0βˆ’π‘Šπ΄ξ€·π‘‘0ξ€Έ.(5.3) Then, (5.2) reads 𝑑𝑣𝑑𝑑+𝐴𝛼𝑣𝑑𝑣=𝐺(𝑣,𝑑),0ξ€Έ=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∢=[𝑑0,𝑑0+𝜏) be an interval in ℝ. We say that a stochastic process 𝑣(𝑑,πœ”;𝑑0,𝑣0) is a mild solution of the system (5.4) in 𝐻𝑠, if 𝑣⋅,πœ”;𝑑0,𝑣0ξ€Έξ€·βˆˆπΆπΌ;𝐻𝑠,𝐏-a.s.πœ”βˆˆΞ©,(5.5) and satisfies in 𝐻𝑠 the following integral equation: 𝑣𝑑,πœ”;𝑑0,𝑣0ξ€Έ=π‘’βˆ’π΄π›Ό(π‘‘βˆ’π‘‘0)𝑣0βˆ’ξ€œπ‘‘π‘‘0π‘’βˆ’π΄π›Ό(π‘‘βˆ’π‘ )ξ€·π΅π›Όβˆ’1𝑓(𝑒)βˆ’π›½π‘Šπ΄(𝑠)𝑑𝑠,𝐏-a.s.πœ”βˆˆΞ©.(5.6)

Theorem 5.2. Let 𝛼=0. Suppose that the Hypothesis (Q2) is satisfied.
Then, for every 𝑒0∈𝐻2, there is a unique globally defined mild solution 𝑣(𝑑,πœ”;𝑑0,𝑣0) of (5.4) in 𝐻2 with 𝑣𝑑,πœ”;𝑑0,𝑣0ξ€Έπ‘‘βˆˆπΆξ€·ξ€Ί0ξ€Έ,∞;𝐻2ξ€Έβˆ©πΆ0,1βˆ’π‘Ÿloc𝑑0,ξ€Έ;𝐻4π‘Ÿξ€Έπ‘‘βˆ©πΆξ€·ξ€·0ξ€Έ,∞;𝐻4ξ€Έ,(5.7) for all 0β‰€π‘Ÿ<1.

Proof. We only consider the case where 𝑛=3. First, it is easy to verify that 𝐏-a.s. 𝐺(𝑣,𝑑)∈𝐢Lip;𝛾𝐻2×𝑑0ξ€Έξ€Έ,∞,𝐻.(5.8) Indeed, by Lemma 3.3, we see that π‘Šπ΄(𝑑)∈𝐻2 is 𝛾-HΓΆlder continuous with respect to π‘‘βˆˆβ„   𝐏-a.s. Recall that 𝑓 is a polynomial of degree 2π‘βˆ’1 with 𝑝=2 (in case 𝑛=3). One deduces that there exist 𝐢1,𝐢2(πœ”)>0 such that ||𝐺𝑣1,𝑑1ξ€Έξ€·π‘£βˆ’πΊ2,𝑑2||≀𝐢1ξ€·||𝑣1βˆ’π‘£2||2+||π‘Šπ΄ξ€·π‘‘1ξ€Έβˆ’π‘Šπ΄ξ€·π‘‘2ξ€Έ||2≀𝐢2ξ€·||𝑣(πœ”)1βˆ’π‘£2||2+||𝑑1βˆ’π‘‘2||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 [𝑑0,𝑇) satisfying 𝐏-a.s. 𝑣𝑑,πœ”;𝑑0,𝑣0ξ€Έ=π‘’βˆ’π΄2(π‘‘βˆ’π‘‘0)𝑣0βˆ’ξ€œπ‘‘π‘‘0π‘’βˆ’π΄2(π‘‘βˆ’π‘ )𝐴𝑓(𝑒(𝑠))βˆ’π›½π‘Šπ΄(𝑣𝑠)𝑑𝑠,𝑑,πœ”;𝑑0,𝑣0ξ€Έπ‘‘βˆˆπΆξ€·ξ€Ί0ξ€Έ,𝑇;𝐻2ξ€Έβˆ©πΆ0,1βˆ’π‘Ÿloc𝑑0ξ€Έ,𝑇;𝐻4π‘Ÿξ€Έπ‘‘βˆ©πΆξ€·ξ€·0ξ€Έ,𝑇;𝐻4ξ€Έ,(5.10) for all 0β‰€π‘Ÿ<1. Furthermore, we also know that 𝑣 is a strong solution in 𝐻2. Hence, it satisfies in the strong sense that 𝑑𝑣𝑑𝑑+𝐴2𝑣+𝐴𝑓(𝑒)βˆ’π›½π‘Šπ΄ξ€·π‘‘=0,𝑣0ξ€Έ=𝑣0.(5.11) In what follows, we show 𝑇=∞, thus proving the theorem.
Simple computations yields ||||≀||𝑓Δ𝑓(𝑒)ξ…ž||(𝑒)𝐿∞||||+||π‘“Ξ”π‘’ξ…žξ…ž||(𝑒)𝐿∞||||βˆ‡π‘’2𝐿4.(5.12) Since 𝑓 is a polynomial of degree 3, there exist πœ…1 and πœ…2 such that ||π‘“ξ…ž||(𝑠)β‰€πœ…1ξ€·1+|𝑠|2ξ€Έ,||π‘“ξ…žξ…ž||(𝑠)β‰€πœ…2(1+|𝑠|),βˆ€π‘ βˆˆβ„.(5.13) Therefore, ||π‘“ξ…ž||(𝑒)𝐿∞||||Ξ”π‘’β‰€πœ…1ξ€·1+|𝑒|2πΏβˆžξ€Έ||||Δ𝑒≀2πœ…1ξ‚€1+|𝑣|2𝐿∞+||π‘Šπ΄||2πΏβˆžξ‚ξ€·||||+||Ξ”π‘£Ξ”π‘Šπ΄||ξ€Έ.(5.14) By the Nirenberg-Gagiardo inequality, there exist 𝐢3,𝐢4,𝐢5>0 such that |𝑒|2πΏβˆžβ‰€πΆ3||||Δ𝑒2,π‘’βˆˆπ»2,|𝑒|2πΏβˆžβ‰€πΆ4||Ξ”2𝑒||1/3|𝑒|𝐿5/36,π‘’βˆˆπ»4,||||Δ𝑒≀𝐢5||Ξ”2𝑒||1/2||||βˆ‡π‘’1/2,π‘’βˆˆπ»4.(5.15) Hence, ||π‘“ξ…ž||(𝑒)𝐿∞||||Δ𝑒≀2πœ…1ξ‚€1+|𝑣|2𝐿∞+||π‘Šπ΄||2πΏβˆžξ‚ξ€·||||+||Ξ”π‘£Ξ”π‘Šπ΄||≀2πœ…1ξ‚€1+𝐢4||Ξ”2𝑣||1/3|𝑣|𝐿5/36+𝐢3||Ξ”π‘Šπ΄||2𝐢5||||βˆ‡π‘£1/2||Ξ”2𝑣||1/2+||Ξ”π‘Šπ΄||.(5.16) By (Q2) and Lemma 3.2, we know that for 𝐏-a.s. πœ”βˆˆΞ©, there exists an 𝑅1(πœ”)>0 such that |Ξ”π‘Šπ΄(𝑑)|≀𝑅1(πœ”) (for all π‘‘βˆˆβ„). 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. πœ”βˆˆΞ©, there exist 𝐢6(πœ”),𝐢7(πœ”)>0 such that |𝑣|𝐿5/36≀𝐢6||||(πœ”),βˆ‡π‘£1/2≀𝐢7(πœ”),(5.17) where the continuous imbedding 𝐻1β†ͺ𝐿6 is used. Consequently, we have ||π‘“ξ…ž||(𝑒)𝐿∞||||Δ𝑒≀𝐢8ξ‚€||Ξ”(πœ”)1+2𝑣||1/3+𝑅1||Ξ”(πœ”)2𝑣||1/2+𝑅1(πœ”),𝐏-a.s.πœ”βˆˆΞ©.(5.18)
Similarly for 𝐏-a.s. πœ”βˆˆΞ©, one easily deduces that there exists 𝐢9(πœ”)>0 such that ||π‘“ξ…žξ…ž||(𝑒)𝐿∞||||Δ𝑒≀𝐢9ξ‚€||Ξ”(πœ”)1+2𝑣||1/6+𝑅1||Ξ”(πœ”)2𝑣||1/4+𝑅1(πœ”).(5.19) It then follows from (5.12) that for 𝐏-a.s. πœ”βˆˆΞ©, ||||Δ𝑓(𝑒)≀𝐢8ξ‚€||Ξ”(πœ”)1+2𝑣||1/3+𝑅1||Ξ”(πœ”)2𝑣||1/2+𝑅1(πœ”)+𝐢9ξ‚€(πœ”)1+𝐿3||Ξ”2𝑣||1/6+𝑅1||Ξ”(πœ”)2𝑣||1/4+𝑅1(πœ”)≀𝐢10ξ‚€||Ξ”(πœ”)1+2𝑣||5/6.(5.20)
Now, taking the 𝐿2 inner-product of equation (5.11) with Ξ”2𝑣, one obtains 12𝑑||||𝑑𝑑Δ𝑣2+||Ξ”2𝑣||2≀||||ξ€œπΊΞ”π‘“(𝑒)Ξ”2||||||||ξ€œπ‘£π‘‘π‘₯+π›½πΊπ‘Šπ΄Ξ”2||||≀1𝑣𝑑π‘₯4||Ξ”2𝑣||2+||||Δ𝑓(𝑒)2+14||Ξ”2𝑣||2+𝛽2||π‘Šπ΄||2≀12||Ξ”2𝑣||2+||||Δ𝑓(𝑒)2+𝛽2πœ†1βˆ’2||Ξ”π‘Šπ΄||2.(5.21) By (5.20), we deduce that 𝐏-a.s. 𝑑||||𝑑𝑑Δ𝑣2+||Ξ”2𝑣||2≀𝐢11ξ‚€||Ξ”(πœ”)1+2𝑣||5/3.(5.22) Furthermore, by Young's inequality and |Ξ”2𝑣|2β‰₯πœ†21|Δ𝑣|2, we know that there exists 𝐢12(πœ”)>0 such that 𝐏-a.s. 𝑑||||𝑑𝑑Δ𝑣2πœ†β‰€βˆ’212||||Δ𝑣2+𝐢12(πœ”).(5.23) Applying the gronwall lemma on (5.23), one gets ||||Δ𝑣2≀2𝐢12(πœ”)πœ†21,𝐏-a.s.πœ”βˆˆΞ©.(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 𝑒0∈𝐻2.

Theorem 5.3. Let π›Όβˆˆ(0,1], and let Hypothesis (Q1) be satisfied. Then, for every 𝑒0∈𝐻1, there is a unique maximally defined mild solution 𝑣(𝑑,πœ”;𝑑0,𝑣0) of (5.4) in 𝐻1 for all π‘‘βˆˆ[𝑑0,∞) with 𝑣𝑑,πœ”;𝑑0,𝑣0ξ€Έπ‘‘βˆˆπΆξ€·ξ€Ί0ξ€Έ,∞;𝐻1ξ€Έβˆ©πΆ0,1βˆ’π‘Ÿloc𝑑0ξ€Έ,∞;𝐻2π‘Ÿξ€Έπ‘‘βˆ©πΆξ€·ξ€·0ξ€Έ,∞;𝐻2ξ€Έ,(5.25) for 0β‰€π‘Ÿ<1.

Proof. As noted above, βˆ’π΄π›Ό is a positive selfadjoint linear operator on 𝐻 with compact resolvent. The negative operator βˆ’π΄π›Ό generate an analytic semigroup π‘’βˆ’π΄π›Όπ‘‘. It is easy to verify by Lemma 3.1 that 𝐏-a.s. 𝐺(𝑣,𝑑)∈𝐢Lip;𝛾𝐻1×𝑑0ξ€Έξ€Έ,∞,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 [𝑑0,𝑇) with 𝑣𝑑,πœ”;𝑑0,𝑣0ξ€Έπ‘‘βˆˆπΆξ€·ξ€Ί0ξ€Έ,𝑇;𝐻1ξ€Έβˆ©πΆ0,1βˆ’π‘Ÿloc𝑑0ξ€Έ,𝑇;𝐻2π‘Ÿξ€Έπ‘‘βˆ©πΆξ€·ξ€·0ξ€Έ,𝑇;𝐻2ξ€Έ,(5.27) where 𝑑0<𝑇=𝑇(𝑣0)β‰€βˆž, and 0β‰€π‘Ÿ<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 𝐡𝛼𝑑𝑣𝑑𝑑+𝐴𝑣=βˆ’π‘“π‘£+π‘Šπ΄ξ€Έ+π›½π΅π›Όπ‘Šπ΄ξ€·π‘‘,𝑣0ξ€Έ=𝑣0.(5.28) Multiplying (5.28) by 𝐴𝑣, one gets 12𝑑𝑑𝑑|𝑣|21,𝐡𝛼+|𝑣|22+𝑓𝑣+π‘Šπ΄ξ€Έξ€Έξ€·π΅,𝐴𝑣+π›½π›Όπ‘Šπ΄ξ€Έ,𝐴𝑣=0.(5.29) We observe that 𝑓𝑣+π‘Šπ΄ξ€Έξ€Έ=ξ€œ,π΄π‘£πΊξ€·βˆ‡π‘“π‘£+π‘Šπ΄ξ€Έ=ξ€œβˆ‡π‘£π‘‘π‘₯πΊπ‘“ξ…žξ€·π‘£+π‘Šπ΄ξ€Έ||||βˆ‡π‘£2ξ€œπ‘‘π‘₯+πΊπ‘“ξ…žξ€·π‘£+π‘Šπ΄ξ€Έβˆ‡π‘Šπ΄βˆ‡π‘£π‘‘π‘₯.(5.30) We take πΆξ…ž1 and πΆξ…ž2 such that π‘“ξ…ž(π‘₯)β‰₯2π‘βˆ’12π‘Ž2π‘βˆ’1π‘₯2π‘βˆ’2βˆ’πΆξ…ž1,||π‘“ξ…ž(||π‘₯)≀2(2π‘βˆ’1)π‘Ž2π‘βˆ’1π‘₯2π‘βˆ’2+𝐢′2,(5.31) for all π‘₯βˆˆβ„. Then, 𝑓𝑣+π‘Šπ΄ξ€Έξ€Έβ‰₯,𝐴𝑣2π‘βˆ’12π‘Ž2π‘βˆ’1ξ€œπΊ||𝑣+π‘Šπ΄||2π‘βˆ’2||||βˆ‡π‘£2𝑑π‘₯βˆ’πΆξ…ž1ξ€œπΊ||||βˆ‡π‘£2𝑑π‘₯βˆ’2(2π‘βˆ’1)π‘Ž2π‘βˆ’1ξ€œπΊ||𝑣+π‘Šπ΄||2π‘βˆ’2||||||βˆ‡π‘£βˆ‡π‘Šπ΄||𝑑π‘₯βˆ’πΆξ…ž2ξ€œπΊ||||||βˆ‡π‘£βˆ‡π‘Šπ΄||β‰₯1𝑑π‘₯4(2π‘βˆ’1)π‘Ž2π‘βˆ’1ξ€œπΊ||𝑣+π‘Šπ΄||2π‘βˆ’2||||βˆ‡π‘£2𝑑π‘₯βˆ’2πΆξ…ž1ξ€œπΊ||||βˆ‡π‘£2𝑑π‘₯βˆ’πΆξ…ž3ξ‚΅ξ€œπΊ||βˆ‡π‘Šπ΄||2π‘ξ€œπ‘‘π‘₯+𝐺||βˆ‡π‘Šπ΄||2ξ‚Ά,𝑑π‘₯(5.32) where we have used HΓΆlder's inequality, Young's inequality, and the appropriate imbeddings 𝐻1(𝐺)β†ͺπΏπ‘Ÿ(𝐺) in dimension 𝑛=1π‘œπ‘Ÿ2 and 3. We also know by (2.6) that there exists 𝛼≀𝐢𝛼≀𝑀1 such that ξ€·π΅π›Όπ‘Šπ΄ξ€Έ,π΄π‘£β‰€πΆπ›Όξ‚΅ξ€œπΊ||βˆ‡π‘Šπ΄||2ξ€œπ‘‘π‘₯+𝐺||||βˆ‡π‘£2𝑑π‘₯.(5.33) Combining the last two inequalities together, we deduce that there exists constants πΆξ…ž4,πΆξ…ž5>0 such that 12𝑑|𝑑𝑑𝑣|21,𝐡𝛼+|𝑣|22+14(2π‘βˆ’1)π‘Ž2π‘βˆ’1ξ€œπΊ||𝑣+π‘Šπ΄||2π‘βˆ’2||||βˆ‡π‘£2𝑑π‘₯≀2πΆξ…ž4ξ€œπΊ||||βˆ‡π‘£2𝑑π‘₯+πΆξ…ž5ξ‚΅ξ€œπΊ||βˆ‡π‘Šπ΄||2π‘ξ€œπ‘‘π‘₯+𝐺||βˆ‡π‘Šπ΄||2ξ‚Ά.𝑑π‘₯(5.34) In view of (2.7), there exists π›Όβ‰€πΆξ…žπ›Όβ‰€π‘€1 such that 12πΆξ…žπ›Όπ‘‘|𝑑𝑑𝑣|21+|𝑣|22+14(2π‘βˆ’1)π‘Ž2π‘βˆ’1ξ€œπΊ||𝑣+π‘Šπ΄||2π‘βˆ’2||||βˆ‡π‘£2𝑑π‘₯≀2πΆξ…ž4|𝑣|21+πΆξ…ž5ξ‚΅ξ€œπΊ||βˆ‡π‘Šπ΄||2π‘ξ€œπ‘‘π‘₯+𝐺||βˆ‡π‘Šπ΄||2ξ‚Ά.𝑑π‘₯(5.35) Using the gronwall lemma on (5.35), the following inequality holds: |𝑣|21≀2𝑒4𝐢′4/𝐢′𝛼𝐢′4||𝑣0||21+2𝑒4𝐢′4/πΆβ€²π›Όξ€œπ‘‘π‘‘0πΆξ…ž5ξ‚΅ξ€œπΊ||βˆ‡π‘Šπ΄(||𝑠)2π‘ξ€œπ‘‘π‘₯+𝐺||βˆ‡π‘Šπ΄(||𝑠)2𝑑π‘₯𝑑𝑠.(5.36) Lemma 3.1 guarantees that 𝐏-a.s. ξ€œπ‘‘π‘‘0ξ€œπΊ||βˆ‡π‘Šπ΄(||𝑠)2π‘ξ€œπ‘‘π‘₯𝑑𝑠<+∞,𝑑𝑑0ξ€œπΊ||βˆ‡π‘Šπ΄(||𝑠)2𝑑π‘₯𝑑𝑠<+∞.(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 𝑒0↦𝑒(𝑑,πœ”;𝑑0,𝑒0) defines a stochastic flow 𝑆𝛼(𝑑,𝑠;πœ”),𝑆𝛼(𝑑,𝑠;πœ”)𝑒0ξ€·=𝑒𝑑,πœ”;𝑠,𝑒0ξ€Έ[],π›Όβˆˆ0,1.(6.1) Notice that 𝐏-a.s. (i)𝑆𝛼(𝑑,𝑠;πœ”)=𝑆𝛼(𝑑,π‘Ÿ;πœ”)βˆ˜π‘†π›Ό(π‘Ÿ,𝑠;πœ”),forallπ‘ β‰€π‘Ÿβ‰€π‘‘,(ii)𝑆0(𝑑,𝑠;πœ”) is continuous in 𝐻2, and 𝑆𝛼(𝑑,𝑠;πœ”) is continuous in 𝐻1 for 0<𝛼≀1.

6.2. Compactness Properties of Stochastic Flow 𝑆𝛼(𝑑,𝑠;πœ”)

Lemma 6.1. (i) Under Assumption (ππŸβˆ—), the stochastic flow 𝑆0(𝑑,𝑠;πœ”) is uniformly compact at time 0. More precisely, for all π΅βŠ‚π»2 bounded and each 𝑑0<0, 𝑆0(0,𝑑0;πœ”)𝐡 is v relatively compact in 𝐻2.
(ii) Under Assumption (ππŸβˆ—), the flow 𝑆𝛼(𝑑,𝑠;πœ”), 0<𝛼≀1, is uniformly compact at time 0. More precisely, for all π΅βŠ‚π»1 bounded and each 𝑑0<0, 𝑆𝛼(0,𝑑0;πœ”)𝐡 is 𝐏-a.s. relatively compact in 𝐻1.

Proof. (i) Let π΅βŠ‚π»2 be a given bounded deterministic set. By Lemma 3.4, we know that for 𝐏-a.s. πœ”βˆˆΞ©, there exists 𝑅2(πœ”)>0, such that |π‘Šπ΄(𝑑)|2+πœŽβ‰€π‘…2(πœ”), π‘‘βˆˆβ„. Define 𝐡=𝐡βˆͺ𝐡2+𝜎(0,𝑅2(πœ”)), where 𝐡2+𝜎(0,𝑅2(πœ”)) denotes the open ball centered at 0 with radius 𝑅2(πœ”) in 𝐻2+𝜎. Then, ξπ΅βŠ‚π»2 is 𝐏-a.s. bounded, and 𝑆0ξ€·0,𝑑0𝑒;πœ”π΅βŠ‚π΄2𝑑0𝑣0βˆ’ξ€œ0𝑑0𝑒𝐴2𝑠𝐺(𝑣(𝑠),𝑠)𝑑𝑠+π‘Šπ΄(0),𝑣0βˆˆξπ΅ξ‚ΌβŠ‚π‘1+𝑁2+𝑁3+𝑁4,(6.2)𝐏-a.s., where 𝑁1=𝑒𝐴2𝑑0𝑁𝐡,2=ξ‚»ξ€œ0βˆ’π›Ώπ‘’π΄2𝑠𝐺(𝑣(𝑠),𝑠)𝑑𝑠,𝑣0βˆˆξπ΅ξ‚Ό,𝑁3=π‘’βˆ’π΄2π›Ώξ‚»ξ€œπ‘‘βˆ’π›Ώ0𝑒𝐴2(𝑠+𝛿)𝐺(𝑣(𝑠),𝑠)𝑑𝑠,𝑣0βˆˆξπ΅ξ‚Ό,𝑁4=𝐡2+πœŽξ€·0,𝑅2ξ€Έ,(πœ”)(6.3) and 𝛿 is an arbitrary constant satisfying 0<𝛿<βˆ’π‘‘0.

Since for 𝑑>0 fixed the operator π‘’βˆ’π΄2𝑑 is compact, we see that 𝑁1, 𝑁3, and 𝑁4 are relatively compact sets in 𝐻2. Now, we show that 𝐏-a.s. 𝑆0(0,𝑑0;πœ”)𝐡 is relatively compact. To this end, we first give an estimate on the Kuratowski measure of 𝑁2βŠ‚π»2.

For 𝑣0∈𝐡, one has ||||ξ€œ0βˆ’π›Ώπ‘’π΄2(π‘ βˆ’π‘‘0)||||𝐺(𝑣(𝑠),𝑠)𝑑𝑠2=||||ξ€œ0βˆ’π›Ώπ΄π‘’π΄2(π‘ βˆ’π‘‘0)||||.𝐺(𝑣(𝑠),𝑠)𝑑𝑠(6.4) Since 𝐴2 is a positive sectorial operator on 𝐻, there exists a constant 𝑀𝐴>0 such that |||π΄π‘’βˆ’π΄2𝑑|||β„’(𝐻2)β‰€π‘€π΄π‘‘βˆ’1/2,βˆ€π‘‘β‰₯0.(6.5) Recall that 𝐺(𝑣,𝑑)∈𝐢Lip;𝛾(𝐻2Γ—[𝑑0,+),𝐻). So there is a 𝐾0(πœ”)>0 such that 𝐏-a.s. ||||𝐺(𝑣,𝑑)≀𝐾0[](πœ”),βˆ€(𝑣,𝑑)βˆˆπ΅Γ—βˆ’π›Ώ,0.(6.6) Therefore ||||ξ€œ0βˆ’π›Ώπ‘’π΄2𝑠||||𝐺(𝑣(𝑠),𝑠)𝑑𝑠2≀𝐾0(πœ”)π‘€π΄ξ€œ0βˆ’π›Ώ(βˆ’π‘ )1/21𝑑𝑠=2𝐾0(πœ”)𝑀𝐴𝛿1/2.(6.7) It follows that πœ…ξ€·π‘2≀diam𝐻2𝑁2≀𝐾0𝑀𝐴𝛿1/2,(6.8) where πœ…(β‹…) 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 πœ…ξ‚€π‘†0ξ€·0,𝑑0𝐡𝑁;πœ”β‰€πœ…1𝑁+πœ…2𝑁+πœ…3𝑁+πœ…4ξ€Έξ€·π‘β‰€πœ…2≀𝐾0𝑀𝐴𝛿1/2.(6.9) Letting 𝛿→0, one immediately concludes that 𝐏-a.s. πœ…(𝑆0(0,𝑑0;πœ”)𝐡)=0, hence 𝑆0(0,𝑑0𝐡;πœ”) is relatively compact.

(ii) The proof of the compactness result for 𝑆𝛼(𝑑,𝑠;πœ”) (0<𝛼≀1) is fully analogous, and is thus omitted.

6.3. The Random Attractors

Now, we show that the system 𝑆𝛼(𝑑,𝑠;πœ”) possesses a random attractor π’œπ›Ό(πœ”) for every π›Όβˆˆ[0,1].

Proof of Theorem 2.4. We infer from the proofs of Theorem 5.2 and Lemma 6.1 that there exists 𝑑(πœ”)<0 such that for any 𝑑0≀𝑑(πœ”), we can define an absorbing set for 𝑆0(𝑑,𝑑0;πœ”) at time 0 by 𝔅0=ξƒ―||||π‘£βˆΆΞ”π‘£2≀2𝐢12(πœ”)πœ†21ξƒ°βˆͺ𝐡2+πœŽξ€·0,𝑅2ξ€Έ(πœ”),(6.10) and for 𝑆𝛼(𝑑,𝑠;πœ”) (0<𝛼≀1), for any 𝑑0<0 we can define an absorbing set for 𝑆𝛼(𝑑,𝑑0;πœ”) at time 0 by 𝔅𝛼=𝐡1(0,Ξ¨),(6.11) where 𝐡1(0,Ξ¨) denotes the open ball centered at 0 with radius Ξ¨ 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 (Ξ©,β„±,𝐏) be a probability space. We consider a family of mappings {𝑆(𝑑,𝑠;πœ”)}𝑑β‰₯𝑠,πœ”βˆˆΞ©βˆΆπ‘‹βŸΆπ‘‹,(A.1) satisfying 𝐏-a.s. (i)𝑆(𝑑,𝑠;πœ”)=𝑆(𝑑,π‘Ÿ;πœ”)βˆ˜π‘†(π‘Ÿ,𝑠;πœ”),forallπ‘ β‰€π‘Ÿβ‰€π‘‘,(ii)𝑆(𝑑,𝑠;πœ”) is continuous in 𝑋, for all 𝑠≀𝑑.

Definition A.1. We say that ℬ(𝑑,πœ”)βŠ‚π‘‹ is an absorbing set at time 𝑑, if 𝐏-a.s. (i)ℬ(𝑑,πœ”) is bounded,(ii)for all π΅βŠ‚π‘‹ there exists 𝑠𝐡 such that 𝑆(𝑑,𝑠;πœ”)π΅βŠ‚β„¬(𝑑,πœ”), for all 𝑠≀𝑠𝐡.

Definition A.2. Given π‘‘βˆˆβ„ and πœ”βˆˆΞ©, we say that {𝑆(𝑑,𝑠;πœ”)}𝑑β‰₯𝑠,πœ”βˆˆΞ© is uniformly compact at time t if for all bounded set π΅βŠ‚π‘‹, there exist 𝑠𝐡, such that 𝐏-a.s. ξšπ‘ β‰€π‘ π΅π‘†(𝑑,𝑠;πœ”)𝐡(A.2) is relatively compact in 𝑋.

Definition A.3. Given π‘‘βˆˆβ„ and πœ”βˆˆΞ©, for any set π΅βŠ‚π‘‹, we define the random omega limit set of a bounded set π΅βŠ‚π‘‹ at time 𝑑 as Ω𝐡(𝑑,πœ”)=π‘‡β‰€π‘‘ξšπ‘ β‰€π‘‡π‘†(𝑑,𝑠;πœ”)𝐡.(A.3)

Definition A.4. Let (𝑋,𝑑) be a metric space, and let {𝑆(𝑑,𝑠;πœ”)}𝑑β‰₯𝑠,πœ”βˆˆΞ© a family of operators that maps 𝑋 into itself. We say that π’œ(𝑑,πœ”) is a stochastic attractor if 𝐏-a.s. (i)π’œ(𝑑,πœ”) is not empty and compact,(ii)𝑆(𝜏,𝑠;πœ”)π’œ(𝑠,πœ”)=π’œ(𝜏,πœ”) for all 𝜏β‰₯𝑠,(iii)for every bounded set π΅βŠ‚π‘‹, limπ‘‘β†’βˆ’βˆžπ‘‘(𝑆(𝑑,𝑠;πœ”)𝐡,π’œ(𝑑,πœ”))=0.

Remark A.5. (i) In the stochastic case, it is not possible to construct the random attractor as the Ξ©-limit of the absorbing set (as done in the deterministic case). This is due to the fact that the Ξ©-limit set is taken from βˆ’βˆž 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 π΅βŠ‚π‘‹ and {𝑆(𝑑,𝑠;𝑀)}𝑑β‰₯𝑠,πœ”βˆˆΞ© is uniformly compact at time 𝑑, then the RDS possesses a random attractor defined by π’œ(𝑑,πœ”)=ξšπ΅βŠ‚π‘‹Ξ©π΅(𝑑,πœ”).(A.4)

Remark A.7. In this paper, we write π’œ(πœ”) instead of π’œ(0,πœ”) for short.


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