Existence of a random attractor for non-autonomous stochastic plate equations with additive noise and nonlinear damping on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{n}$\end{document}

Based on the abstract theory of pullback attractors of non-autonomous non-compact dynamical systems by differential equations with both dependent-time deterministic and stochastic forcing terms, introduced by Wang in (J. Differ. Equ. 253:1544–1583, 2012), we investigate the existence of pullback attractors for the non-autonomous stochastic plate equations with additive noise and nonlinear damping on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{n}$\end{document}.


Introduction
Plate equations have been studied for many years because of their worth in certain physical areas such as vibration and elasticity theories of solid mechanics. The research of the longtime dynamical behavior of plate equations has become an important area in the field of the infinite-dimensional dynamical system. The purpose of this paper is to investigate the following non-autonomous stochastic plate equations with additive noise and nonlinear damping defined in the entire space R n : with the initial value conditions As we know, the attractor is regarded as a proper notation describing the long-time dynamics of solutions, and many classical literature works and monographs have appeared for both the deterministic and stochastic dynamical systems over the last decades, see [1,5,7,8,10,11,14,19,23,25] and the references therein. However, in reality, a system is always affected by some random factors such as external noise. In order to scrutinize the large-time behavior and characterization of solution for the stochastic partial differential equations driven by noise, Crauel and Flandoi [7,8], Flandoi and Schmalfuss [10], and Schmalfuss [19] introduced the concept of pullback attractors and established some abstract results for the existence of such attractors about compact dynamical system [1,8,10,14,15]. Since these methods required the compactness of a pullback absorbing set for systems, they could not be used to deal with the stochastic PDEs on unbounded domains. Therefore, in [3], Bates, Lisei, and Lu presented the concept of asymptotic compactness for random dynamical systems, which is an extension of deterministic systems. And then, using these abstract results, they proved the existence of random attractors for reaction-diffusion equations on unbounded domain in [4]. Wang in [25] further extended the concept of asymptotic compactness to the case of partial differential equations with both random and time-dependent forcing terms; moreover, he applied these criteria into the stochastic reaction-diffusion equation with additive noise on R n and obtained the existence of a unique pullback attractor. For most of works on stochastic PDEs, please refer to [9,22,[27][28][29]32] and the references therein.
Just for problem (1.1)-(1.2) and the corresponding plate equations, in the deterministic case (i.e., ε = 0), existence of global attractors has been studied by several authors, see for instance [2, 12-14, 30, 31, 33, 34, 37]. As far as the stochastic case driven by additive noise goes, when the deterministic forcing term g is independent of time, that is, g(x, t) ≡ g(x), the existence of a random pullback attractor on bounded domain has been obtained in [17,20,21]. Recently, on the unbounded domain, the authors investigated the existence and upper semi-continuity of random attractors for stochastic plate equation with rotational inertia and Kelvin-Voigt dissipative term as well as dependent-on-time terms (see [36] for details) and asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains [35]. To the best of our knowledge, it has not been considered by any predecessors for the stochastic plate equation with additive noise and nonlinear damping on unbounded domain. It is well known that nonlinear damping makes the problem more complex and interesting even to the case of bounded domain. Besides, the theory and applications of Wang in [24][25][26] gave us the idea of solving this problem and inspired us, so we decided to study the existence of pullback attractors for problem (1.1)-(1.2).
Notice that (1.1) is a non-autonomous stochastic equation, i.e., the external term g is time-dependent. In this case, as in [25], we introduce two parametric spaces to present its dynamics: one is for the deterministic non-autonomous perturbations, while the other for the stochastic perturbations. In addition, since Sobolev embeddings are not compact on R n , we cannot get the asymptotic compactness directly from the regularity of solutions. We conquer the difficulty by using the uniform estimates on the tails of solutions outside a bounded ball in R n and the splitting technique [27] and the compactness methods introduced in [16].
The organization of this paper is as follows: In Sect. 2, we present some notations and a proposition about random dynamical systems. In Sect. 3, we establish a continuous cocycle for Eq. (1.1) in H 2 (R n ) × L 2 (R n ). In Sect. 4, we obtain all necessary uniform estimates of solutions. Finally, in Sect. 5, we show the existence and uniqueness of a random attractor for (1.1)-(1.2), denoted by R n .
Throughout the paper, the letters c and c i (i = 1, 2, . . .) are positive constants which may change their values from line to line or even in the same line.

Preliminaries
In order to state and prove our main results, we introduce some notations and a proposition related to random attractors for stochastic dynamical systems.
Let X be a separable Banach space and (Ω, F, P) be the standard probability space, where Ω = {ω ∈ C(R, R) : ω(0) = 0}, F is the Borel σ -algebra induced by the compact open topology of Ω, and P is the Wiener measure on (Ω, F). There is a classical group {θ t } t∈R acting on (Ω, F, P) which is defined by (2.1) We often say that (Ω, F, P, {θ t } t∈R ) is a parametric dynamical system.

Definition 2.2 ([6])
Assume that Φ is a continuous cocycle on X over R and (Ω, F, P, {θ t } t∈R ), and D is the collection of all tempered families of nonempty bounded subsets of X parameterized by τ ∈ R and ω ∈ Ω:

Definition 2.3 ([6]
) D is said to be tempered if there exists x 0 ∈ X such that, for every c > 0, τ ∈ R, and ω ∈ Ω, the following holds:
Definition 2.6 ([6]) A family K = {K(τ , ω) : τ ∈ R, ω ∈ Ω} ∈ D is called a D-pullback absorbing set for Φ if, for all τ ∈ R and ω ∈ Ω and for every D ∈ D, there exists T = T(D, τ , ω) > 0 such that is closed in X and is measurable in ω with respect to F .

Proposition 2.1 ([25])
Suppose that Φ is D-pullback asymptotically compact in X and has a closed measurable D-pullback absorbing set K in D. Then Φ has a unique D-pullback attractor A in D which is given by, for each τ ∈ R and ω ∈ Ω, (2.10)

Cocycles for stochastic plate equation
In this section, we firstly present the precise hypotheses on problem (1.1)-(1.2), then show that it generates a continuous cocycle in H 2 (R n ) × H 1 (R n ). 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: As usual, (·, ·) denotes L 2 -inner product and · denotes the L 2 -norm. Let E = H 2 × L 2 , with the Sobolev norm Let ξ = u t + δu, where δ is a small positive constant whose value will be determined later, then (1.1)-(1.2) can be reduced to the equivalent system with the initial value conditions Assume that the functions h, f satisfy the following conditions: where β > 0, 1 ≤ p ≤ n+4 n-4 . Note that (3.4) and (3.5) imply (2) There exist two constants β 1 , β 2 such that To study the dynamical behavior of problem (3.2)-(3.3), we need to convert the stochastic system into a deterministic one with a random parameter. To this end, we set v(t) = ξ (t)φω(t), we obtain the equivalent system of (3.2)-(3.3): (3.10) with the initial value conditions By a standard method as in [5,18,23,36], one may show the following lemma under conditions (3.4)-(3.9).
also generates a continuous cocycle with (3.2)-(3.3) over R and (Ω, F, P, {θ t } t∈R ). Note that these two continuous cocycles are equivalent. By (3.13), it is easy to check that Φ has a random attractor provided Φ possesses a random attractor. Then we only need to consider the continuous cocycle Φ.
Next we make another assumption: Assume that σ , δ, and g satisfy the following conditions: and where | · | denotes the absolute value of a real number in R.

Given a bounded nonempty subset
Let D be the collection of all such families, that is,

Uniform estimates of solutions
In this section, we derive uniform estimates on the solutions of the stochastic plate equa- It is easy to check that · E is equivalent to the usual norm · H 2 ×L 2 in (3.1). The next lemma shows that the cocycle Φ has a pullback D-absorbing set in D.
and R 1 (τ , ω) is given by where M is a positive constant independent of τ , ω, D.
Proof Taking the inner product of the second equation of (3.10) with v in L 2 (R n ), we find that By the first equation of (3.10), we have By Lagrange's mean value theorem and (3.9), we get where ϑ is between 0 and v + φω(t)δu. By (3.9) and (4.4), we get Then substituting the v in (4.4) into the third and fourth terms on the left-hand side of (4.3), we find that Using the Cauchy-Schwarz inequality and Young's inequality, we have Then, for the last term on the left-hand side of (4.3), we have (4.11) By condition (3.5) we get Using condition (3.4) and (3.6), we obtain By (4.11)-(4.13), we get (4.14) Substitute (4.5)-(4.14) into (4.3) to obtain Let σ = min{δ, δc 2 2 }, then Multiplying (4.16) by e σ t and then integrating over (τt, τ ), we have Replacing ω by θ -τ ω in the above, we obtain, for every t ∈ R + , τ ∈ R, and ω ∈ Ω, Again, by (3.9), we get Thus, for the first term on the right-hand side of (4.17), we have Since (u 0 , v 0 ) ∈ D(τt, θ -t ω) and D ∈ D, then we find For the last term on the right-hand side of (4.17), we find Notice that ω(s) has at most linear growth at |s| → ∞, which combines (3.19), we can have Finally, we estimate the fourth term on the left-hand side of (4.17). Thanks to (3.6), we obtain that, for all t ≥ 0, Thus the proof is completed.
and R 2 (τ , ω) is given by Proof Taking the inner product of the second equation of (3.10) with A 1 2 v in L 2 (R n ), we find that Similar to the proof of Lemma 4.1, we have the following estimates: For the last term on the left-hand side of (4.24), thanks to (3.7), we have Plugging (4.25)-(4.31) into (4.24) and together with (3.15), we obtain (4.33) Multiplying (4.33) by e σ t and then integrating over (τt, τ ), we have Replacing ω by θ -τ ω in the above, we obtain, for every t ∈ R + , τ ∈ R, and ω ∈ Ω, where for k ≥ 1, B k = {x ∈ R n : |x| ≤ k} and R n \ B k is the complement of B k .
Proof Choose a smooth function ρ such that 0 ≤ ρ ≤ 1 for s ∈ R, and and there exist constants Taking the inner product of the second equation of (3.10) with Similar to (4.5), we have Taking (4.38) into (4.37), we have For the fourth term on the left-hand side of (4.39), we have For the third term on the left-hand side of (4.39), we have Similar to (4.12) and (4.13) in Lemma 4.1, we have By (4.38)-(4.45), we have we have that, for all k ≥ K 1 , Multiplying (4.49) by e σ t and then integrating over (τt, τ ), we find Replacing ω by θ -τ ω, it then follows from (4.50) that Note that (3.6) holds, then we find from which along with η 3 ∈ L 1 (R n ), we see that there exists Then from (4.53)-(4.54) we know that there exists T 2 = T 2 (τ , ω, D, η) > T 1 such that, for all t ≥ T 2 and k ≥ K 3 , which completes the proof.
Let ρ = 1ρ with ρ given by (4.36). Fix k ≥ 1, and set (4.56) By (3.10)-(3.11) we find that u and v satisfy the following system in B 2k = {x ∈ R n : |x| < 2k}: with boundary conditions Let {e n } ∞ n=1 be an orthonormal basis of L 2 (B 2k ) such that Ae n = λ n e n with zero boundary condition in B 2k . Given n, let X n = span{e 1 , . . . , e n } and P n : L 2 (B 2k ) → X n be the projection operator.

Random attractors
In this section, we prove the existence of a D-attractor for the stochastic system (3.10)-(3.11). We firstly apply the lemmas shown in Sect. 4 to derive the asymptotic compactness of solutions of (3.10)-(3.11).