Asymptotic behavior of plate equations with memory driven by colored noise on unbounded domains

: The paper investigates mainly the asymptotic behavior of the non-autonomous random dynamical systems generated by the plate equations with memory driven by colored noise deﬁned on R n . Firstly, we prove the well-posedness of the equation in the natural energy space. Secondly, we deﬁne a continuous cocycle associated with the solution operator. Finally, we establish the existence and uniqueness of random attractors of the equation by the uniform tail-ends estimates methods and the splitting technique.


Introduction
The colored noise was first introduced in [23,24] in order to obtain the information of velocity of randomly moving particles, which cannot be obtained from the white noise since the the Wiener process is nowhere differentiable. Moreover, for many physical systems, the stochastic fluctuations are correlated and should be modeled by the colored noise rather than the white noise, see [20]. This paper is concerned the asymptotic behavior of the plate equation driven by nonlinear colored noise in unbounded domains: u(x, τ) = u 0 (x), u t (x, τ) = u 1,0 (x), x ∈ R n , t ≤ τ, (1.1) where τ ∈ R, α, ν are positive constants, µ is the memory kernel, f and h are given nonlinearity, g ∈ L 2 loc (R, H 1 (R n )), and ζ δ is a colored noise with correlation time δ > 0.
It is clear that (1.1) becomes a deterministic plate equation as µ ≡ 0 and h ≡ 0. In this case, we can characterize the long-time behavior of solutions by virtue of the concept of global attractors under the framework of semigroup. Some authors have extensively studied the existence of global attractors for the autonomous plate equation. For instance, the attractors of deterministic plate equations have been investigated in [2,8,12,14,30,[32][33][34][35]44] in bounded domains. In [2,30,34,35], the authors considered global attractor for the plate equation with thermal memory; Khanmamedov investigated a global attractor for the plate equation with displacement-dependent damping in [8]; Liu and Ma obtained the existence of time-dependent strong pullback attractors for non-autonomous plate equations in [12,14]; Yang and Zhong studied the uniform attractor and global attractor for non-autonomous plate equations with nonlinear damping in [32,33], respectively; In [44], the author obtained global existence and blow-up of solutions for a Kirchhoff type plate equation with damping. For the case of unbounded domains, see refereces [9,10,13,31,42].
The existence and uniqueness of pathwise random attractors of stochastic plate equations have been studied in [15,16,21,22] in the case of bounded domains; and in [36][37][38][39][40][41] in the case of unbounded domains. In all these publications ( [36][37][38][39][40][41]), only the additive white noise and linear multiplicative white noise were considered. Notice that the random equation (1.1) is driven by the colored noise rather than the white noise. In general, it is very hard to study the asymptotic dynamics of differential equations driven by nonlinear white noise, including the random attractors. Indeed, only when the white noise is linear, the stochastic equations can be transformed into a deterministic equations, then one can obtain the existence of random attractors of the plate equation (1.1). However, this transformation does not apply to stochastic equations driven by nonlinear white noise, and that is why we are currently unable to prove the existence of random attractors for systems with nonlinear white noise.
For the colored noise, even it is nonlinear, we are able to show system (1.1) has a random attractor in H 2 (R n ) × L 2 (R n ) × R µ,2 (the definition of R µ,2 see Section 3), which is quite different from the nonlinear white noise. The reader is referred to [6,7,26,27] for more details on random attractors of differential equations driven by colored noise. However, for the random plate equations driven by colored noise (1.1), we find that there is no results available to the existence of random attractors. In the present paper, we will prove that (1.1) is pathwise well-posed and generate a continuous cocycle, and the cocycle possesses a unique tempered random attractor. This is different from the corresponding stochastic system driven by white noise where the symbol • indicates that the equation is understood in the sense of stratonovich integration. For (1.2), one can define a random dynamical system when h(·, ·, u) is a linear function, see [41]. But for a general nonlinear function h, random dynamical system associated with (1.2) can not be defined due to the absence of appropriate transformation, hence asymptotic behavior of such stochastic equations has not been investigated until now by the random dynamical system approach. This paper indicates that the colored noise is much easier to handle than the white noise for studying pathwise dynamics of such stochastic equations. The main purpose of the paper is establish the existence and uniqueness of measurable tempered random attractors in H 2 (R n ) × L 2 (R n ) × R µ,2 for the dynamical system associated with (1.1). The key for achieving our goal is to establish the tempered pullback asymptotic compactness of solutions of (1.1) in H 2 (R n ) × L 2 (R n ) × R µ,2 . Involving to our problem (1.1), there are two essential difficulties in verifying the compactness. On the one hand, notice that system (1.1) is defined in the unbounded domain R n where the noncompactness of Sobolev embeddings on unbounded domains gives rise to difficulty in showing the pullback asymptotic compactness of solutions, to get through of it, we use the tail-estimates method (as in [25]) and the splitting technique (see [3]) to obtain the pullback asymptotic compactness. On the other hand, there is no applicable compact embedding property in the "history" space. In this case, we solve it with the help of a useful result in [19]. For our purpose, we introduce a new variable and an extend Hilbert space.
The rest of this article consists of four sections. In the next section, we define some functions sets and recall some useful results. In Section 3, we first establish the existence, uniqueness and continuity of solutions in initial data of (1.1) in H 2 (R n ) × L 2 (R n ) × R µ,2 , then define a non-autonomous random dynamical system based on the solution operator of problem (1.1). The last two section are devoted to derive necessary estimates of solutions of (1.1) and the existence of random attractors.
Throughout the paper, the inner product and the norm of L 2 (R n ) will be denoted by (·, ·) and || · ||, respectively. The letters c and c i (i = 1, 2, . . .) are generic positive constants which may depend on some parameters in the contexts.

Asymptotic compactness of cocycles
In this section, we define some functions sets and recall some useful results, see [4,17,18,28,29,43]. These results will be used to establish the asymptotic compactness of the solutions and attractor for the random plate equation defined on the entire space R n .
From now on, we assume (Ω, F , P) is the canonical probability space where Ω = {ω ∈ C(R, R) : ω(0) = 0} with compact-open topology, F is the Borel σ-algebra of Ω, and P is the Wiener measure on (Ω, F ). Recall the standard group of transformations {θ t } t∈R on Ω: Suppose Φ : R + × R × Ω × X → X is a continuous cocycle on X over (Ω, F , P, {θ t } t∈R ). Let D be a collection of some families of nonempty subset of X: Suppose Φ has a D-pullback absorbing set K = {K(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D; that is, for every τ ∈ R, ω ∈ Ω and D ∈ D there exists T = T (τ, ω, D) > 0 such that for all t ≥ T , Assume that where both Φ 1 and Φ 2 are mappings from R + × R × Ω × X to X.
Given k ∈ N, denote by O k = {x ∈ R n : |x| < k} andÕ k = {x ∈ R n : |x| > k}. Let X be a Banach space with norm · X which consists of some functions defined on R n . Given a function u : R n → R, the restrictions of u to O k andÕ k are written as u| O k and u|Õ k , respectively. Denote by Suppose X O k and XÕ k are Banach spaces with norm · O k and · Õ k , respectively, and We further assume that for every δ > 0, τ ∈ R, and ω ∈ Ω, there exists t 0 = t 0 (δ, τ, ω, K) > 0 and k 0 = k 0 (δ, τ, ω) ≥ 1 such that and In addition, we assume that for every k ∈ N, t ∈ R + , τ ∈ R, and ω ∈ Ω, the set Theorem 2.1 [29]. If (2.1)-(2.6) hold, then the cocycle Φ is D-pullback asymptotically compact in X; that is, the sequence {Φ(t n , τ−t n , θ −t n ω, x n )} ∞ n=1 is precompact in X for any τ ∈ R, ω ∈ Ω, D ∈ D, t n → ∞ monotonically, and x n ∈ D(τ − t n , θ −t n ω). Theorem 2.2 [29]. Let D be an inclusion closed collection of some families of nonempty subsets of X, and Φ be a continuous cocycle on X over (Ω, F , P, {θ t } t∈R ). Then Φ has a unique D-pullback random attractor A in D if Φ is D-pullback asymptotically compact in X and Φ has a closed measurable D-pullback absorbing set K in D.

Cocycles of random plate equations
In this section, we first establish the existence of solution for problem (1.1), then define a nonautonomous cocycle of (1.1).
Given δ > 0, let ζ δ (θ t ω) be the unique stationary solution of the stochastic equation: where W is a two-sided real-valued Wiener process on (Ω, F , P). The process ζ δ (θ t ω) is called the one-dimensional colored noise. Recall that there exists a θ t -invariant subset of full measure (see [1]), which is still denoted by Ω, such that for all ω ∈ Ω, ζ δ (θ t ω) is continuous in t ∈ R and lim t→±∞ ζ δ (θ t ω) t = 0.
Let −∆ denote the Laplace operator in R n , A = ∆ 2 with the domain D(A) = H 4 (R n ). We can also define the powers A ν of A for ν ∈ R. The space V ν = D(A ν 4 ) is a Hilbert space with the following inner product and norm Following Dafermos [5], we introduce a Hilbert "history" space R µ,2 = L 2 µ (R + , V 2 ) with the inner product and new variables By differentiation we have Then (1.1) can be rewritten as the equivalent system We introduce the following hypotheses to complete the uniform estimates. Assume that the memory kernel function µ ∈ C 1 (R + ) ∩ L 1 (R + ), and satisfy the following conditions: ∀ s ∈ R + and some > 0.
Proof. The proof will be divided into four steps. We first construct a sequence of approximate solutions, and then derive uniform estimates, in the last two steps we take the limit of those approximate solutions to prove the uniqueness of solutions.
Step (i): Approximate solutions. Given k ∈ N, define a function η k : R → R by (3.14) Then for every fixed k ∈ N, the function η k as defined by (3.14) is bounded and Lipschitz continuous; more precisely, for all s, s 1 , For all x ∈ R n and t, s ∈ R, denote By (3.4) we know that there exists k 0 ∈ N such that for all |s| ≥ k 0 and x ∈ R n , f (x, s)s > 0, (3.17) thus, for all k ≥ k 0 and x ∈ R n , By (3.5)-(3.6), (3.15)-(3.16) and (3.18) we know that for all s, s 1 , s 2 ∈ R and x ∈ R n , and By (3.19) we get that for all s ∈ N and x ∈ R n , By (3.7)-(3.8) and (3.15)-(3.16) we obtain that for all k ≥ 1, t, s, s 1 , s 2 ∈ R and x ∈ R n , (3.23) For every k ∈ N, consider the following approximate system for u k , η t k : (3.26) From (3.23)-(3.24), ϕ ∈ L ∞ (R n ) and the standard method (see, e.g., [11]), it follows that for each τ ∈ R, ω ∈ Ω, u 0 ∈ H 2 (R n ), u 1,0 ∈ L 2 (R n ) and η 0 ∈ R µ,2 , problem (3.26) has a unique global solution (u k , ∂ t u k , η t k ) defined on [τ, τ + T ] for every T > 0 in the sense of Definition 3.1. In particular, for almost all t ∈ [τ, τ + T ]. Next, we use the energy equation (3.25) to derive uniform estimate on the sequence {u k , ∂ t u k , η t k } ∞ k=1 .

Uniform estimates of solutions
In this section, we derive necessary estimates of solutions of (3.2) under stronger conditions than (3.4)-(3.8) on the nonlinear functions f and h. These estimates are useful for proving the asymptotic compactness of the solutions and the existence of pullback random attractors.
For the second term on the right-hand side of (4.14) we get (4.15) By (4.14)-(4.15) we get Multiplying (4.14) by e 1 4 εγt , and then integrating the inequality [τ − t, τ], after replacing ω by θ −τ ω, we get For the first term on the right-hand side of (4.17), by (4.4) we get By (4.7) we get which along with (4.2) and (4.18) that for all t ≥ T , Then the proof is completed. Based on Lemma 4.1, we can easily obtain the following Lemma that implies the existence of tempered random absorbing sets of Φ.  In order to derive the uniform tail-estimates of the solutions of (3.2) for large space variables when times is large enough, we need to derive the regularity of the solutions in a space higher than H 2 (R n ). Lemma 4.3. Let (3.3)-(3.5), (3.8), (4.1)-(4.2) and (4.5)-(4.8) hold. Then for any τ ∈ R, ω ∈ Ω and D ∈ D, there exists T = T (τ, ω, D) > 0 such that for all t ≥ T , the solution of (3.2) satisfies where (u 0 , u 1,0 , η 0 ) ∈ D(τ − t, θ −τ ω) and M 2 is a positive number independent of τ, ω and D.
Proof. Taking the inner product of (3.2) 1 with A Taking the inner product of (1.1) 1 with A 1 2 u t in L 2 (R n ), we find that By (4.20) and (4.21), we get (g(t) + h(t, ·, u(t))ζ δ (θ t ω), εA where the definition of L see Lemma 4.2, and C is the positive constant satisfying , which can be rewritten as For the last term on the right-hand side of (4.27) we have which together with (4.27), we get Similar to the remainder of Lemma 4.1, we can obtain the desired result.

Existence of random attractors
In this section, we present the existence and uniqueness of D-pullback random attractors of (3.2). Let z = (u, u t , η t ) be the solution of (3.2). Denote u =ṽ + v, η t =η t + η where (ṽ,η t ) and (v, η t ) are the solutions of the following equations, respectively, where (u 0 , u 1,0 ) ∈ D(τ − t, θ −t ω) and M 2 is a positive number independent of τ, ω and D.
which together with the invariance of A 1 , we know that the D-pullback random attractor A 1 is a singleton. This complete the proof.
6. The discussion of the proposed method's theoretical analysis In this paper, we use the uniform estimates on the tails of solutions and the splitting technique to obtained the existence and uniqueness of D-pullback attractor for the problem (1.1). The method used in this paper is proposed by P. W. Bates et al [3], they applied the method to deal with the asymptotic behavior of the non-automatous random system on unbounded domains. More precisely, one first need to show that the tails of the solutions of (1.1) are uniformly small outside a bounded domain for large time, and then derive the asymptotic compactness of solutions in bounded domains by splitting the solutions as two parts: one part has trivial dynamics in the sense that it possesses a unique tempered attracting random solution; and the other part has regularity higher than H 2 (R n ) × L 2 (R n ) × R µ,2 based on the estimates of solutions (see Lemma 4.3).

Conclusions
Using the uniform estimates on the tails of solutions and the splitting technique, we obtained the existence and uniqueness of D-pullback attractor for the problem (1.1). It is well-known that the pullback random attractors are employed to describe long-time behavior for an non-autonomous dynamical system with random term, while the D-pullback attractor that we obtained can characterize the asymptotic behavior of the equation like (1.1), which is featured with both stochastic term and non-autonomous term.