Random attractors for non-autonomous stochastic wave equations with nonlinear damping and white noise

This paper is concerned with the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with additive white noise, for which the nonlinear damping has a critical cubic growth rate. By showing the pullback asymptotic compactness of the stochastic dynamic systems, we prove the existence of a random attractor in H1 0 × L2.

(H 2 ) Let f (u, x) = f 1 (u, x) + f 2 (u, x) and F i = u 0 f i (r, x) dr, where f 1 (·, x) ∈ C 2 (R, R), f 2 (·, x) ∈ C 1 (R, R). Furthermore, f 1 , f 2 meet conditions that there exist constants c 1 , c 2 , c 3 , c 4 > 0 and functions φ i (x) ∈ L 1 (U), i = 1, 2, such that In the deterministic damped wave equation (i.e., a = 0), global attractors have been studied by many authors, such as [1][2][3] and the reference therein. In addition, uniform attractors and pullback attractors also attracted many experts' attention, cf. [4][5][6][7][8]. If the function g does not depend on time, (1)-(3) is an autonomous stochastic wave equation, and its random attractors have been explored in [9][10][11][12][13]. For many problems, such as wave propagation through the atmosphere or the ocean, the more realistic models must take the random fluctuation into account. So it is important and interesting to study random attractors. For non-autonomous random dynamical systems, Wang established an efficacious theory about the existence of random attractors [14][15][16][17]. Particularly, Li [18] studied the asymptotic dynamics for a stochastic damped wave equation with multiplicative noise defined on unbounded domains and proved the existence of random attractors. For the non-autonomous stochastic strongly damped wave equation, the existence of random attractors is proved in [19][20][21]. Lv and Wang [10] also studied the existence of random attractors for the stochastic wave equation and showed the upper semicontinuous dependence of the random attractor on parameters. The authors in [22] studied the asymptotic behavior of a class of non-autonomous nonlocal fractional stochastic parabolic equations driven by multiplicative white noise on the entire space R n .
In this paper, (1)-(3) is a non-autonomous stochastic system where the external term g is time-dependent. We shall transform the stochastic wave equation into a deterministic one with random parameter and random initial data through an Ornstein-Uhlenbeck process z(θ t ω), then prove the existence of a random attractor for the random dynamical system generated by (1)-(3). It is well known that the key step in proving the existence of attractors in both deterministic and random systems is to establish the compactness of the system in some sense. Motivated by [23], we will work out this problem.
The paper is arranged as follows. In Sect. 2, we collect some basic concepts and background material about random attractor for the random dynamical system generated by (1)-(3), then the existence and uniqueness of solutions is established in Sect. 3. In Sect. 4, we consider the concrete bounds of the solution and decompose the solution of (12)- (13) into two parts. In Sect. 5, we establish the asymptotic compactness of the random dynamical system and obtain the existence of the random attractor.

Random dynamical systems
In this section, we collect some basic definitions and known results about general random dynamical systems (see [17,24,25] for details).
Let (Ω, F, P) be a probability space, where Ω = {ω ∈ C(R, R) : ω(0) = 0} is endowed with compact-open topology. F is the Borel σ -algebra on Ω and P is the corresponding Wiener measure on F . For any t, let (θ t ) t∈R on Ω via thus (Ω, F, P, (θ t ) t∈R ) is an ergodic metric dynamical system [24]. In the following, X labels as a Banach or Hilbert space with the Borel σ -algebra B(X). Definition 2.1 Let {θ t } t∈R be a family of (B(R × F), F)-measurable mappings, θ t : R × Ω → Ω such that θ 0 (·) is the identity on Ω, θ s+t (·) = θ t (·) • θ s (·) for all t, s ∈ R and Pθ t = P for all t ∈ R. Definition 2.2 Let (Ω, F, P, (θ t ) t∈R ) be a parametric dynamical system. 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 (i)-(iv) are satisfied: is a (usually closed) nonempty subset of X and the mapping ω ∈ Ω → d(x, D(τ , ω)) is (F , B(R))-measurable for every fixed x ∈ X and τ ∈ R, then D = D(τ , ω) : τ ∈ R, ω ∈ Ω is called a random set. (2) Let D be a collection of random sets in a Polish space X. A continuous cocycle Φ is said to be pullback D-asymptotically compact (D-a.c.) in X if, for any τ ∈ R, ω ∈ Ω, D ∈ D and any sequences t n → +∞, x N ∈ D(τt n , θ -t n ω), the sequence Φ(t n , τt n , θ -t n ω, x n ) has a convergent subsequence in X.
Then K is called a pullback D-absorbing set for Φ if, for all τ ∈ R, ω ∈ Ω and for every D ∈ D, there exists t 0 (K, τ , ω) > 0 such that and ω ∈ Ω. In addition, if there exists T > 0 such that A(τ + T, ω) = A(τ , ω) for any τ ∈ R, ω ∈ Ω, then A is periodic with period T. Proposition 2.1 Let D be a neighborhood-closed collection of (τ , ω)-parametrized families of nonempty subsets of X and Φ be a continuous cocycle on X over R and (Ω, F, P, {θ t } t∈R ). Then Φ has a pullback D-attractor A in D if and only if Φ is pullback D-asymptotically compact in X and Φ has a closed F -measurable pullback D-absorbing set K in D. The unique pullback D-attractor A = A(τ , ω) is given by

Proposition 2.2 Let D be a neighborhood-closed collection of (τ , ω)-parametrized families of nonempty subsets of X. If Φ is a continuous τ -periodic cocycle with period T > 0 on X over R and (Ω, F, P, {θ t } t∈R ) and if Φ has a pullback D-attractor A ∈ D, then A is τ -periodic with period T if and only
with its inner product and norm as follows: The letters c and c i (i = 1, 2, . . . ) are generic positive constants which do not depend on ω, τ , t, a.

Existence and uniqueness of solutions
In this section, motivated by [26,27], we establish the existence and uniqueness of solutions for Eqs. (1)-(3). Let λ be the first eigenvalue of the operator A := -on U with Dirichlet boundary conditions. Note that A : In the following, we convert problem (1)-(3) into a random system without noise terms. Identify ω(t) with W (t), i.e., ω(t) = W (t), t ∈ R, and let z(θ t ω) := -0 -∞ e s (θ t ω)(s) ds (t ∈ R) be a Ornstein-Uhlenbeck stationary process which solves the Itô equation dz + z dx = dW (t).
(1)-(3) can be rewritten aṡ Thus where Since R ε,θ t ω : (a, b) T → (a, b + εaah(x)z(θ t ω)) T is an isomorphism of E, then Φ, Γ , Ψ are equivalent to each other in dynamics. Therefore, the existence of random attractors in any of these stochastic dynamical systems means that random attractors also exist in other dynamical systems. We will consider the existence of a random attractor for RDS Φ in the following.
For the component ϕ N , which is ultimately pullback bounded in a higher regular space, we have the following estimate.
The proof is completed.