Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains

Abstract We study the asymptotic behavior of solutions to the non-autonomous stochastic plate equation driven by additive noise defined on unbounded domains. We first prove the uniform estimates of solutions, and then establish the existence and upper semicontinuity of random attractors.


Introduction
Consider the following non-autonomous stochastic plate equation with additive noise de ned in the entire space R n : with the initial value conditions u(x, τ) = u (x), u t (x, τ) = u (x), where x ∈ R n , t > τ with τ ∈ R, α > , λ > and β are constants, f is a nonlinearity satisfying certain growth and dissipative conditions, g(x, ·) and h are given functions in L loc (R, L (R n )) and H (R n ), respectively, W(t) is a two-sided real-valued Wiener process on a probability space. Plate equations have been investigated for many years due to their importance in some physical areas such as vibration and elasticity theories of solid mechanics. The study of the long-time dynamics of plate equations has become an outstanding area in the eld of the in nite-dimensional dynamical system. While the attractors are regarded as a proper notation to describe the long-time dynamics of solutions. To the best of our knowledge, there have been many works on the investigation of the attractors for the plate equations over the last few years. For instance, if the random term is vanished and g(x, t) = g(x), then (1.1)-(1.2) change into a deterministic autonomous plate equation. The existence and uniqueness of the global attractor of the corresponding dynamical system was studied in [1][2][3][4][5][6][7][8][9][10]; besides, the uniform attractor of the dynamical system generated by the non-autonomous plate equation was established in [11].
For the stochastic plate equations, if the forcing term g(x, t) = g(x), then the existence of a random attractor of (1.1)-(1.2) on bounded domain have been proved in [12][13][14]. However, it is not yet considered by any predecessors to the stochastic plate equation on unbounded domain. In recent years, the existence of random attractors for stochastic dynamical system on unbounded domains have been investigated by several authors, such as Reaction-di usion equations with additive noise [15], Reaction-di usion equations with multiplicative noise [16], FitzHugh-Nagumo equations with additive noise [17,18], Navier-Stokes equations with additive noise [19], wave equations with additive noise [20][21][22], wave equations with multiplicative noise [23].
Motivated by above literatures, the goal of the present paper is to study random attractors and its upper semicontinuity of non-autonomous stochastic equation (1.1). By applying the abstract results in [24][25][26], we will rst prove the stochastic plate equation (1.1) has tempered random attractors in H (R n ) × L (R n ), then establish the upper semicontinuity of the random attractors.
In general, the existence of global random attractor depends on some kind compactness (see, e.g., [27][28][29][30]). To prove the existence of random attractors for (1.1) in H (R n ) × L (R n ), we must establish the pullback asymptotic compactness of solutions. Since Sobolev embeddings are not compact on unbounded domain, we cannot get the desired asymptotic compactness directly from the regularity of solutions. We here overcome the di culty by using the uniform estimates on the tails of solutions outside a bounded ball in R n and the splitting technique, see [20,31] for details.
The framework of this paper is as follows. In the next Section, we recall a su cient and necessary criterion for existence of pullback attractors for cocycle or nonautonomous random dynamical systems. In Section 3, we de ne a continuous cocycle for Eq. (1.1) in H (R n )×L (R n ). Then we derive all necessary uniform estimates of solutions in Section 4. In Section 5, we prove the existence and uniqueness of tempered random attractor for the non-autonomous stochastic plate equation. Finally, in Section 6, we prove the upper semicontinuity of random attractors as β to zero.
Throughout the paper, the letters c and c i (i = , , . . .) are generic positive constants which may change their values from line to line or even in the same line.

Preliminaries
In this section, we recall some basic concepts 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( ) = }, 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 de ned by ( . ) We often say that (Ω, F, P, {θ t } t∈R ) is a parametric dynamical system. The following four de nitions and one proposition are from [24].
De nition 2.1. A mapping Φ : R + × R × Ω × X → X is called a continuous cocycle on X over R and (Ω, F, P, {θ t } t∈R ) if for all τ ∈ R, ω ∈ Ω and t, s ∈ R + , the following conditions (1)-(4) are satis ed: Hereafter, we assume 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: D is said to be tempered if there exists x ∈ X such that for every c > , τ ∈ R and ω ∈ Ω, the following holds: lim Given D ∈ D, the family Ω(D) = {Ω(D, τ, ω) : τ ∈ R, ω ∈ Ω} is called the Ω-limit set of D where The cocycle Φ is said to be D-pullback asymptotically compact in X if for all τ ∈ R and ω ∈ Ω, the sequence De nition 2.2. 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) > such that If, in addition, K(τ, ω) is closed in X and is measurable in ω with respect to F, then K is called a closed measurable D-pullback absorbing set for Φ.

De nition 2.3. A family
where d H is the Hausdor semi-distance given by d H (F, G) = sup u∈F inf v∈G ||u − v|| X , for any F, G ⊂ X.
As in the deterministic case, random complete solutions can be used to characterized the structure of a D-pullback attractor. The de nition of such solutions are given below.
Proposition 2.1. Suppose Φ is D-pullback asymptotically compact in X and has a closed measurable Dpullback 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,

Cocycles for stochastic plate equation
In this section, we outline some basic settings about (1.1)-(1.2) and show that it generates a continuous cocycle in H (R n ) × L (R n ).
Let −∆ denote the Laplace operator in R n , A = ∆ with the domain D(A) = H (R n ). We can de ne the powers A ν of A for ν ∈ R. The space Vν = D(A ν ) is a Hilbert space with the following inner product and norm For brevity, the notation (·, ·) for L -inner product will also be used for the notation of duality pairing between dual spaces. We denote H = H (R n ) × L (R n ). We de ne a new norm · H by for Y = (u, v) ∈ H, where stands for the transposition. 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 rewritten as the equivalent system with the initial value conditions where The function f will be assumed to satisfy the following conditions: ( . ) We also need the following condition on g: there exists a positive constants σ such that where | · | denotes the absolute value of real number in R. For our purpose, it is convenient to convert the problem (1.1)-(1.2) (or (3.2)-(3.3)) into a deterministic system with a random parameter, and then show that it generates a cocycle over R and (Ω, F, P, {θ t } t∈R ).
We identify ω(t) with W(t), i.e., ω(t) = W(t) = W(t, x), t ∈ R. Consider Ornstein-Uhlenbeck equation dy + ydt = dW(t), and Ornstein-Uhlenbeck process From [32], it is known that the random variable |y(ω)| is tempered, and there is a θ t -invariant set Ω ⊂ Ω of full P measure such that y(θ t ω) is continuous in t for every ω ∈ Ω. Put which solves dz + zdt = hdW . ( . ) Lemma 3.1 [33] For any ε > , there exists a tempered random variable γ : with the initial value conditions where We will consider (3.9)-(3.10) for ω ∈ Ω and write Ω as Ω from now on. The well-posedness of the deterministic problem (3.9)-(3.10) in H (R n )×L (R n ) can be established by standard methods as in [34,35] generates a continuous cocycle with (3.2)-(3.3) over over R and (Ω, F, P, {θ t } t∈R ). Note that these two continuous cocycles are equivalent. By (3.12), it is easy to check that Φ β has a random attractor provided Φ β possesses a random attractor. Then, we only need to consider the continuous cocycle Φ β .
One can prove Φ β is measurable by using the same method as in the following paper: H. Cui, J.A. Langa, Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Di er. Equ. 30 (2018) 1873-1898.

Uniform estimates of solutions
In this subsection, we derive uniform estimates on the solutions of the stochastic plate equations (3.9)-(3.10) de ned on R n when t → ∞. These estimates are necessary for proving the existence of bounded absorbing sets and the asymptotic compactness of the random dynamical system associated with the equations. In particular, we will show that the tails of the solutions for large space variables are uniformly small when time is su ciently large.
Let δ ∈ ( , ) be small enough such that and de ne σ appearing in (3.5) by where c is the positive constant in (F2).
Proof. Taking the inner product of the second equation of (3.9) with v in L (R n ), we nd that By the rst equation of (3.9), we have Then substituting the above v into the second, third and last terms on the right-hand side of (4.3), we nd that From condition (F2) we get By conditions (F1) and (F3), it yields Using the Cauchy-Schwartz inequality and the Young inequality, there holds Recalling the norm · H in (3.1). By (4.1) we obtain from (4.13) that (4.14) Multiplying (4.14) by e σt and then integrating over (τ − t, τ), we have Substituting ω by θ−τ ω, then we have from (4.15) that  (τ, ω) : τ ∈ R, ω ∈ Ω} is a closed measurable D-pullback absorbing set for the continuous cocycle associated with problem (3.9)- (3.10) in D, that is, for every τ ∈ R, ω ∈ Ω, and D = {D(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D, there exists T = T(τ, ω, D) > , such that for all t ≥ T, Proof. This is an immediate consequence of (3.11) and Lemma 4.1.
Choose a smooth function ρ, such that ≤ ρ ≤ for s ∈ R, and and there exist constants Given r ≥ , denote Hr = {x ∈ R n : |x| < r} and R n \ Hr the complement of Hr. To prove asymptotic compactness of solution on R n , we prove the following lemma.
Proof. Note that the cocycle Φ β is pullback D-asymptotically compact in H(R n ) by Lemma 5.1. On the other hand, the cocycle Φ β has a pullback D-absorbing set by Lemma 4.1. Then the existence and uniqueness of a pullback D-attractor of Φ β follow from Proposition 2.1 immediately.

Upper semicontinuity of pullback attractors
First, we present a criteria concerning upper semicontinuity of non-autonomous random attractors with respect to a parameter in [23]. Theorem 6.1 Let (X, · X ) be a separable Banach space and Φ be an autonomous dynamical system with the global attractor A in X. Given β > , suppose that Φ β is the perturbed random dynamical system with a random attractor A β ∈ D and a random absorbing set E β ∈ D. Then for P − a.e. τ ∈ R, ω ∈ Ω, if the following conditions are satis ed: (i) there exists some deterministic constant c such that, for P − a.e. τ ∈ R, ω ∈ Ω lim sup β→ E β (τ, ω) X ≤ c.
(iii) for P − a.e. τ ∈ R, ω ∈ Ω, t ≥ , βn → , and xn , x ∈ X with xn → x, it holds that Next, we will use Theorem 6.1 to consider an upper semicontinuity of random attractors A β (ω) when β → . To indicate the dependence of solutions on β, we respectively write the solutions of problem (3.9)-(3.10) as u (β) and v (β) , that is, (u (β) , v (β) ) satis es When β = , the random problem (3.9)-(3.10) reduces to a deterministic one: Accordingly, by Theorem 5.1 the deterministic and autonomous system Φ generated by (6.2) is readily veri ed to admit a global attractor A in H(R n ).
Authors' contributions: All authors contributed equally to the writing of this paper. All authors read and approved the nal manuscript.