EXISTENCE AND UPPER SEMI-CONTINUITY OF RANDOM ATTRACTORS FOR NON-AUTONOMOUS STOCHASTIC PLATE EQUATIONS WITH MULTIPLICATIVE NOISE ON R N ∗

Based on the abstract theory of random attractors of non-autonomous non-compact dynamical systems, we investigate existence and the upper semi-continuity of random attractors for the non-autonomous stochastic plate equations with multiplicative noise defined on the entire space R n . We extend and improve the results of [42] not only from the additive white noise to the multiplicative white noise, but also from the time-independent of forcing term g ( x ) to the time-dependent forcing term g ( x, t ) .

The study of plate equations have been paid extensive attention to by some of the researches due to their importance in both the physical and engineering areas such as vibration and elasticity theory of solid mechanics; besides, the long-time dynamics of solutions associated with this problem has also located to an important position and become more and more outstanding in the field of the infinite-dimensional dynamical systems .
As we know, attractor is a proper concept describing the long-time dynamics of the dynamical systems, and there are many classical literatures and monographs not only for the deterministic but also for the stochastic dynamical systems over the last two decades years, see for instance( [1, 2, 6-12, 14-19, 22, 23, 26, 27, 30-32, 36, 41]) and references therein. In order to scrutinize the asymptotic behavior of solution for the stochastic partial differential equations driven by noise, H. Crauel & F. Flandoi( [11,12]), F. Flandoi & B. Schmalfuss( [15]) and B. Schmalfuss( [27]) et al. introduced a concept of pullback attractors respectively, and established some abstract results proving existence of such attractors( [1,12,15,22]). However, the compactness of pullback absorbing set was necessary for obtaining the existence of random attractors if we exploit above mentioned methods, so it could not be used to deal with the stochastic PDEs on unbounded domains. In order to implement such defects, P. W. Bates, H. Lisei & K. Lu presented the concept of asymptotic compactness, and applied this technique into the lattic dynamical systems( [5]) and the reaction-diffusion equations on unbounded domain( [4]), respectively. B. Wang in [32] further developed the concept of asymptotic compactness, and obtained existence of a unique pullback attractor for the stochastic reaction-diffusion equations with additive noise on R n . As far as the corresponding other works on stochastic PDEs, we refer to ( [13,14,30,[33][34][35][36][37][38]41]) and references therein.
Only for our problem (1.1)-(1.2), under the deterministic case (i.e., ε = 0), existence of global attractors has been studied by several authors, see for instance [3, 19-21, 39, 40, 43] and reference therein. As far as the stochastic case driven by additive noise, when the forcing term g is independent of time, that is, g(x, t) ≡ g(x), existence of random pullback attractors on bounded domain was obtained in [24,28,29]. Recently, X. Yao, Q. Ma & T. Liu investigated existence and upper semi-continuity of random attractors for stochastic plate equations with rotational inertia and Kelvin-Voigt dissipative term on an unbounded domain(see [42] for details). To the best of our knowledge, it is not yet considered by any predecessors for the stochastic plate equations with multiplicative noise, so we focus on this problem on unbounded domain in the present paper. It is well known that the multiplicative noise makes the problem more complex and interesting even to the environment of bounded domain. Motivated by the theory and applications of B. Wang in [32,36,38], and also based on the works of the stochastic plate equations with rotational inertia and Kelvin-Voight dissipative term on unbounded domains by Yao, Ma & Liu, we are concerned with the existence and upper semi-continuity of random attractors for problem (1.1)-(1.2).
Notice that (1.1) is a non-autonomous stochastic differential equation with the time-dependent external forcing term g, like in [32], we need to introduce two parametric spaces so that describe its dynamics: one is responsible for the deterministic non-autonomous perturbations, and another for the stochastic perturbations. On the other hand, since Sobolev embeddings are not compact on unbounded domain, we can not get the desired asymptotic compactness directly via the regularity of solutions. In order to move these obstacles, we take advantage of the uniform estimates on the tails of solutions outside a bounded ball in R n and along with the splitting technique( [33]), as well as the compactness methods(that is so called "C-Condition" or "flattening Condition") introduced in [17,18].
The remainder of this paper is as follows. In the next Section, we recall a suffi-cient and necessary conditions proving existence of random attractors for cocycle or non-autonomous random dynamical systems. In Section 3, we define a continuous cocycle for (1.1) in H 2 (R n ) × L 2 (R n ) under the condition that insure the wellposedness of solutions. Then we derive all uniform estimates of solutions in Section 4, and prove the existence of random attractors in Sections 5, Finally, we further show the upper semi-continuity of random attractors in the last Section. Throughout the paper, we use || · || and (·, ·) to denote the norm and the inner product of L 2 (R n ), respectively. The norms of L p (R n ) and a Banach space X are generally written as || · || p and || · || X . The letters c and c i (i = 1, 2, . . .) are generic positive constants which may change their values from line to line or even in the same line and do not depend on ε.

Preliminaries
In this section, we recall some definitions and known results regarding pullback attractors of non-autonomous random dynamical systems from ( [1,10,11,32]) which they are useful to our problem.
In the sequel, we use (Ω, F, P) and (X, d) to denote a probability space and a complete separable metric space, respectively. If A and B are two nonempty subsets of X, then we use d(A, B) to denote their Hausdorff semi-distance.
Then the family {Ω(B, τ, ω) : τ ∈ R, ω ∈ Ω} is called the Ω-limit set of B and is denoted by Ω(B). Definition 2.7. Let D be a collection of some families of nonempty bounded subsets of X and K = {K(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D. Then K is called a D-pullback absorbing set for Φ if for all τ ∈ R and ω ∈ Ω and for every B ∈ D, there exists 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. Definition 2.8. Let D be a collection of some families of nonempty bounded subsets of X. Then Φ is said to be D-pullback asymptotically compact in X if for all τ ∈ R and ω ∈ Ω, the sequence Definition 2.9. Let D be a collection of some families of nonempty bounded subsets of X and A = {A(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D. Then A is called a D-pullback attractor for Φ if the following conditions (1)-(3) are fulfilled: that is, for all t ∈ R + , τ ∈ R and ω ∈ Ω, Proposition 2.1. Let D be an inclusion-closed collection of some families of nonempty bounded subsets of X, and Φ be a continuous cocycle on X over R and (Ω, F, P, {θ t } t∈R ). If Φ is D-pullback asymptotically compact in X and Φ 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,

Cocycles associated with 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 We define a new norm ∥ · ∥ H by where ⊤ stands for the transposition. Let z = u t + δu, where δ is a small positive constant whose value will be determined later. Substituting u t = z − δu into (1.1) we find with the initial value conditions ds for x ∈ R n and u ∈ R. The function f ∈ C 2 (R n × R, R) will be assumed to satisfy the following conditions: We also need the following condition on g like in [32]: there exists a positive constant σ such that where γ is a given number by (F1), which implies that where | · | denotes the absolute value of real number in R. Since γ ≥ 1, by (3.6) it is easy to see that for every τ ∈ R, For our purpose, let (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 Then (Ω, F, P, {θ t } t∈R ) is a parametric dynamical system. Now we convert the problem (3.2)-(3.4) into a deterministic system with a random parameter. To this end, we consider the Ornstein-Uhlenbeck equation dy(θ t ω) + εy(θ t ω)dt = dw(t), and Ornstein-Uhlenbeck process From [1,14,22], 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.

Uniform estimates of solutions
In this subsection, we derive uniform estimates on the solutions of problem (3.10)-(3.12) defined 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 sufficiently large. We first obtain the estimates in H.
where M and c are positive constants depending on λ, σ, α and δ, but independent of τ, ω, D and ε. In addition, the random variable R has the property: Proof. Taking the inner product of (3.11) with v in L 2 (R n ), we find that Then substituting the above v into the third, fourth and last terms on the left-hand side of (4.3), there holds It follows from (4.3)-(4.7) that For the last term on the right-hand side of (4.8), by (F1) and (F3), we have where c depends on In line with Young's inequality, we find (4.10) Then by (F2), (4.8)-(4.10) we get (4.11) According to (F3) we know F (x, u) + ϕ 3 (x) ≥ 0 for all x ∈ R n and u ∈ R. This along with (3.16) implies that is, Thus, combining with (4.11) and (4.12), it leads to Recalling the norm ∥ · ∥ H in (3.1), we obtain from (4.13) that Multiplying (4.14) by e ∫ t 0 (2σ−εc−εc|y(θrω)| 2 )dr and then integrating over Replacing ω by θ −τ ω in the above inequality we claim that for every t ∈ R + , τ ∈ R, and ω ∈ Ω, Since |y(θ t ω)| is stationary and ergodic, we get from (3.9) and the ergodic theorem (see [14] for details) that where c is the positive number in (4.16). By virtue of (4.18)-(4.19), we have |y(θ t ω)| is tempered, so integer with (3.7) and (4.20) we conclude that the following integral Therefore, it follows from (4.16), (4.21) and (4.22) that there exists T2 = T2(τ, ω, D) ≥ T1 such that for all t ≥ T 2 , which along with (F3) implies that Next, we prove R 1 (τ, ω) has the following property: for every τ ∈ R, ω ∈ Ω, Collecting all (4.18)-(4.21) we get, for every t ≥ T 1 , τ ∈ R and ω ∈ Ω, which along with (3.6) implies (4.25).
The following estimates are used to prove pullback asymptotic compactness of solutions.
Proof. Multiplying (4.14) by e ∫ t 0 (2σ−εc−εc|y(θrω)| 2 )dr and then integrating over On one hand, for all t ≥ T 1 , s ∈ [−t, 0], we have the following estimates for the last term on the right-hand side of (4.27): where R 2 (τ, ω) is given by On the other hand, there exists which along with (F3) complete the proof. Next, we establish uniform smallness of solutions for large space and time variables. These estimates are important for proving asymptotic compactness of solutions on the unbounded domain R n . For simplicity, we denote Q k = {x ∈ R n : |x| ≤ k} and R n \ Q k the complement of Q k in the sequel.
We now derive uniform estimates of solutions in bounded domains. These estimates will be used to establish pullback asymptotic compactness. Let ρ = 1 − ρ with ρ given by (4.33). Fix k ≥ 1, and set (4.55) By (3.10)-(3.12) we find that u and v satisfy the following system in Q 2k = {x ∈ R n : |x| ≤ 2k}: with boundary conditions Let {e n } ∞ n=1 be an orthonormal basis of L 2 (Q 2k ) such that ∆ 2 e n = λ n e n with boundary condition (4.58) in Q 2k . Given n, let X n = span{e 1 , · · · , e n } and P n : L 2 (Q 2k ) → X n be the projection operator.

Random attractors
In this section, we shall prove the existence of a D-pullback attractor for the random system (3.10)-(3.12) by using Proposition 2.1. First we apply the lemmas shown in Section 4 to prove pullback asymptotic compactness of solutions of (3.10)- (3.12) in Lemma 5.1. Assume that (F1)-(F4) and (3.6) hold. Then for all τ ∈ R and ω ∈ Ω, the solution sequence of (3.10)-(3.12), Proof. According to Lemma 4.1 and the assumption t m → ∞, we see that, for every τ ∈ R and ω ∈ Ω, there exists m 1 = m 1 (τ, ω, D) > 0 such for all m ≥ m 1 , In line with Lemma 4.3 , we find that for every η > 0, there exist k 0 = k 0 (τ, ω, η) ≥ 1 and m 2 = m 2 (τ, ω, D, η) ≥ m 1 and such that for all m ≥ m 2 , Let u and v be the functions defined by (4.55). Then by Lemma 4.4, there are On the other hand, due to (5.1) we obtain for all m ≥ m 3 , (4.55) and the fact that ρ( |x| 2 k 2 ) = 1 for |x| ≤ 1. So together with (5.2) we get the precompactness of the sequence in H 2 (R n ) × L 2 (R n ). Proof. Note that Φ is D-pullback asymptotically compact in H 2 (R n ) × L 2 (R n ) by Lemma 5.1 , and has a closed measurable D-pullback absorbing set by Lemma 4.1 . Therefore, the existence and uniqueness of D-pullback attractor of Φ follows from Proposition 2.1 immediately.

Upper semicontinuity of pullback attractors
In this section, we will consider the upper semicontinuity of pullback attractors for the stochastic plate equation (3.10)-(3.12) on R n . As the critical exponent of f (x, u) is n+4 n−4 , we can't derive the upper semicontinuity of pullback attractors, so we must supplement the following additional condition (this condition has nothing to do with the proof of pullback attractors): where the constant l > 0. First, we present a criteria concerning the upper semicontinuity of non-autonomous random attractors with respect to a parameter in [38].