Random attractors for stochastic retarded strongly damped wave equations with additive noise on bounded domains

Abstract In this paper we study the asymptotic behavior for a class of stochastic retarded strongly damped wave equation with additive noise on a bounded smooth domain in ℝd. We get the existence of the random attractor for the random dynamical systems associated with the equation.


Introduction
The main aim of this paper is to investigate the asymptotic behavior of the solution to the following stochastic strongly damped wave equation with time-delay and with additive noise on a bounded set D ⊂ R d : with the initial value conditions and the boundary condition u(t, x) = , for t ∈ [−h, ∞), x ∈ ∂D.
Here λ and α are positive constants, W j (j = , , · · · , m) is a real-valued two-sided Wiener process on a probability space (Ω, F , P), which will be speci ed later, g is a given function de ned on D, u is a real function, f is a nonlinear functional satisfying some conditions which will be speci ed later, u t (·) = u(t + ·). Wave equation is a kind of hyperbolic equation, which can be used to describe the wave phenomena in nature and engineering. Hence, wave equation is a very important research eld. Some evolution systems in physics, chemistry and life science, depend not only on the current status, but also on the status in the past period. These systems can be described by time-delay partial di erential equations.
In this paper we study the asymptotic behavior of the solution to the stochastic time-delay wave equation (1), when time tends to in nite. As we know, the asymptotic behavior of random system can be studied by

Preliminaries and Random Dynamical Systems
In this section, we rst recall a result for the existence of random attractor for a continuous random dynamical system (RDS), and then introduce some notations which will be used in this paper. At last, we show that the equation (1) generates a random dynamical system.
Let (X, || · || X ) be a Banach space with Borel σ−algebra B(X). Suppose that (Ω, F , P, (θ t ) t∈R ) is a metric dynamical system on the probability space (Ω, F , P). Suppose that ϕ is a continuous RDS of X over (Ω, F , P, (θ t ) t∈R ). And suppose D be a collection of subsets of X. The reader can refer to [3] [7] for more basic knowledge about random dynamical systems. Now, we refer to [3] [13] for the following result for the existence of random attractor for continuous RDS.
This result will be used to prove the existence of random attractor for the RDS generating by stochastic strongly damped wave equation with time-delay (1).
The following notation will be used in the rest of the paper. We use ·, · and || · || to denote the inner product and the norm in L (D), and use the notation || · || X to denote the norm of a general Banach space ||ξ (s)||. (3) Let S be the collection of all continuous function ξ : [−h, ] → L (D) with ||ξ || S < ∞. Then, it is easy to check that (S, || · || S ) is a Banach space. In the following, let (Ω, F , P) be a probability space, with The Borel σ-algebra F on Ω is generated by the compact open topology [1] and P is the corresponding Wiener measure on F . (θ t ) t∈R on Ω is de ned by Then, (Ω, F , P, (θ t ) t∈R ) is an ergodic metric dynamical system. In the rest of this section, we show that there is a RDS generated by the following stochastic strongly damped wave equation: with the initial value conditions and the boundary condition Here α, λ and h are positive constants, g is a given function in L (D), and h j s (j = , , · · · , m) are given functions in H (D), W(t) ≡ (W (t), W (t), · · · , Wm(t)) is a two-sided Wiener process on the probability space (Ω, F , P). We identify ω(t) with W(t), i.e. W j (t) = ω j (t) (j = , , · · · , m). f : S → L (D) is a continuous functional satisfying the following conditions: (A1) f ( ) = ; (A2) there exists a constant L f > such that, for all ξ , η ∈ S, (A3) there exist positive constants β > , Let σ > be a xed constant such that For convenience, we introduce the following natations. Set Set H = H (D) × L (D), and the norm on H is the following Set with the norm ||(u, v)|| H = β ||u|| S + β ||∇u|| S + ||v|| S . Set η = u t + σu, then (6)- (8) can be rewritten as the following form: with the initial value conditions and the boundary condition To show that the equation (15) generates a continuous RDS, we rst transform (15) into a deterministic equation with random parameters. We use famous Ornstein-Uhlenbeck process to do that. For j = , , · · · , m, set It is an Ornstein-Uhlenbeck process, and it is the solution of the following Itô equation: Moreover, the random variable z j (θ t ω j ) is tempered, and z j (θ t ω j ) is P-a.e. continuous [3]. Set then (19) implies that Notice that h j ∈ H (D) (j = , , · · · , m). Thus, there exists a constant c > , such that, It follows from Proposition 4.3.3 [1] that, there exists a tempered function r(ω) > such that where r(ω) satis es, for P-a.e. ω ∈ Ω, Here β is a positive constant which will be xed later. Then, it follows that, for P-a.e. ω ∈ Ω, t ∈ R ||z(θ t ω)|| + ||∇z(θ t ω)|| + ||∆z(θ t ω)|| ≤ ce β |t| r(ω). Set Then the equation (15) can be rewritten as the following form with the initial conditions and the boundary condition The equation (26) is a deterministic equation with random parameters. By [6], under the conditions (A1)-(A3), for each (u , v ) ∈ H, (26) has a unique solution (u(t, ω, u ), v(t, ω, v )) for a.e. ω ∈ Ω. For any T > , by the regularity of solutions for an analytic semigroup [4], we can get that The global solution can be obtained by the boundedness of solution of (26), by Lemma 3.1 below. Hence, equation (26) generates a continuous random dynamical system ϕ with Notice Then, ψ also generates a continuous RDS associate with (15). It is easy to see that two random dynamical systems are equivalent. Thus, we need only to consider the random dynamical system ϕ.

Uniform estimates of solutions
In this section we prove some uniform estimates for the solution of the equation (26). To show that the RDS ϕ has an absorbing set, we need the following assumption: By condition (32), there exist two positive constants β , β , such that Throughout the rest of this paper we assume that D is the collection of all tempered random subsets of H.
Hence, for any B ∈ D, there exists a random variable T (B, ω) > such that, for all t > T (B, ω), It is easy to see that r (ω) is tempered. This ends the proof.