Limiting Behavior of Random Attractors of Stochastic Supercritical Wave Equations Driven by Multiplicative Noise

This paper deals with the limiting behavior of random attractors of stochastic wave equations with supercritical drift driven by linear multiplicative white noise defined on unbounded domains. We first establish the uniform Strichartz estimates of the solutions with respect to noise intensity, and then prove the convergence of the solutions of the stochastic equations with respect to initial data as well as noise intensity. To overcome the non-compactness of Sobolev embeddings on unbounded domains, we first utilize the uniform tail-ends estimates to truncate the solutions in a bounded domain and then employ a spectral decomposition to establish the pre-compactness of the collection of all random attractors. We finally prove the upper semicontinuity of random attractor as noise intensity approaches zero.


Introduction
In this paper, we investigate the stability of pullback random attractors of the stochastic supercritical wave equation driven by multiplicative noise defined on R n : with initial data where 1 ≤ n ≤ 6, τ ∈ R, α and ν are positive constants, f : R n × R → R is a nonlinear term, g ∈ L 2 loc (R, L 2 (R n )), σ ∈ (0, 1) is a parameter representing the noise intensity, the symbol • stands for Stratonovich's integration, and W (t) is a standard real-valued Wiener process on a probability space ( , F , P).
The long term dynamics of the stochastic wave equation (1.1) depends on the growth rate p of the nonlinear term f (x, u) as |u| → ∞. In general, p = n n−2 for n > 2 is called a critical exponent, and the equation is said to be subcritical, critical and supercritical when p < n n−2 , p = n n−2 and p > n n−2 , respectively. In the subcritical or critical case, the nonlinear function f maps H 1 (O) into L 2 (O) for a domain O in R n , which plays a key role for studying the random attractors of the stochastic wave equation, see, e.g., [11,18,32,47,48] for bounded domains and [40,42,43,46,49] for unbounded domains.
However, in the supercritical case, the nonlinear function f does not map H 1 (O) into L 2 (O) any more, and the uniform Strichartz estimates must be used to study the random attractors in this case, see, e.g., [12,13,44,45]. In particular, the existence of random attractors of system (1.1)-(1.2) with supercritical nonlinearity has been proved in [13] recently. In the present paper, we continue this line of research and further investigate the stability of these random attractors for supercritical stochastic wave equation driven by multiplicative noise as the intensity of noise σ → 0.
More precisely, we will prove the random attractors of (1.1)-(1.2) are upper semicontinuous at σ = 0. To that end, we first establish the Strichartz estimates of the solutions which are uniform with respect to noise intensity (see Lemma 3.1). We then prove the pathwise convergence of the solutions of the stochastic equation as σ → 0 (see Lemma 3.2). The main difficulty of the paper is to show the precompactness of the collection of all random attractors of (1.1)- (1.2) in H 1 (R n ) × L 2 (R n ). To overcome the non-compactness of Sobolev embeddings on unbounded domains, we first utilize the invariant property of random attractors as well as the uniform tail-estimates of the solutions to prove that all functions in random attractors are uniformly infinitesimal outside a large bounded domain. We then decompose the solution operator as two parts: one is linear and the other is nonlinear. We prove the linear part is convergent in H 1 (R n ) × L 2 (R n ) as σ → 0. For the nonlinear part in bounded domains, we use the spectral decomposition of the Laplace operator to further split the solutions as a sum of a finite-dimensional component and an infinite-dimensional component.
The paper is organized as follows. In Sect. 2, we review the existence and uniqueness of random attractors of (1.1)-(1.2). In Sect. 3, we prove the convergence of the solutions of (1.1)-(1.2) with respect to initial data and noise intensity. Section 4 is devoted to the precompactness of the collection of all random attractors of (1.
In the last section, we prove the upper semicontinuity of random attractors as the noise intensity σ approaches 0.
Hereafter, we denote the inner product and the norm of L 2 (R n ) by (·, ·) and · , respectively.

Preliminaries
In this section, we review existence of random attractors of the stochastic supercritical wave equation (1.1)-(1.2) on R n , which is needed for further proving the upper semicontinuity of these random attractors.
In the sequel, we assume f : R n × R → R is continuous and write F(x, r ) = r 0 f (x, s)ds for all x ∈ R n and r ∈ R. Suppose f and F satisfy the conditions: for all x ∈ R n and u, u 1 , u 2 ∈ R, where p ≥ 1 for n = 1, 2 and 1 ≤ p < n+2 n−2 for 3 ≤ n ≤ 6, α 1 , We mention that if f (x, u) = |u| p−1 u for x ∈ R n and u ∈ R, then f satisfies all conditions (2.1)-(2.6) for p ≥ 1.
For the probability space, we denote by ( , F , P) the classical Wiener space, where = {ω : R → R is continuous, ω(0) = 0}. Given t ∈ R, denote by Let y(θ t ω) be the unique stationary solution of linear stochastic differential equation Then there exists θ t -invariant set˜ ⊆ with P(˜ ) = 1 such that y(θ t ω) is tempered and continuous in t for any ω ∈˜ , For convenience,˜ will be written as .
Denote by V (t) = u(t)e −σ t 0 y(θ s ω)ds . Then equation (1.1) can be rewritten as follows: with initial data 0 y(θ s ω)ds . As usual, a solution of (2.7)-(2.8) will be understood in the following sense.
is called a solution of (2.7) and (2.8) if We recall from [13] the following existence and uniqueness of solutions to (2.7)-(2.8).
Since both V σ and u satisfy the Strichartz estimates, we see Then by [44,Lemma 3.4] we find that for all t ∈ [τ, τ + T ], n n−2 < p < n+2 n−2 and where C 2 is a positive constant depending on T and ω, but independent of τ, σ and initial data.
In the next section, we prove the uniform precompactness of the family of random attractors {A σ } in H 1 (R n ) × L 2 (R n ).

Uniform Precompactness of Random Attractors
To prove the precompactness of the set {A σ } σ ∈(0,σ 0 ) , we introduce a family of subsets of H 1 (R n ) × L 2 (R n ): for every τ ∈ R and ω ∈ , setting and M 1 is the same number as in (2.18).
For the second term on the right-hand side of (4.42), we have for l > k ≥ k 0 , It follows from (4.23), (4.38), (4.39) and (4.45) that Step 2: prove for every τ ∈ R, k ∈ N and ω ∈ , The idea of the proof is to first truncate the solutions in a bounded domain and then use the spectral decomposition of the Laplace operator with homogeneous Dirichlet boundary condition.
This completes the proof.