Asymptotic Behavior of Stochastic Strongly Damped Wave Equation with Multiplicative Noise

In this paper we study the asymptotic dynamics of the stochastic strongly damped wave equation with multiplicative noise under homogeneous Dirichlet boundary condition. We investigate the existence of a compact random attractor for the random dynamical system associated with the equation.


Introduction
Consider the following stochastic strongly damped wave equation with multiplicative noise: and the initial value conditions ( ) ( ) ( ) ( ) ( ) ( ) where ∆ is the Laplacian with respect to the variable x U ∈ , n U ⊂  is a bounded open set with a smooth boundary U ∂ ; ( ) , u u x t = is a real function of x U ∈ and 0 t ≥ ; 0, c ≠ 0 α > are strong damping coefficients;  denotes the Stratonovich sense of the stochastic term; ( ) is a given external force; ( ) ′ are uniformly bounded and there exist 0 1 2 , , 0 c c c ≥ such that ( ) ( ) where ⋅ denotes the absolute value of number in  .

( )
W t is a one-dimensional two-sided real-valued Wiener process on probability space ( ) , where the Borel σ -algebra  on Ω is generated by the compact open topology, and  is the corresponding Wiener measure on  .We identify ( )

α =
Equation (1.1) can be regarded as a stochastic perturbed model of a continuous Josephson junction [1], which is stochastic damped sine-Gordon equation [2].
A large amount of studies have been carried out toward the dynamics of a variety of systems related to Equation (1.1).For example, the asymptotical behavior of solutions for deterministic and stochastic wave equations has been studied by many authors, see, e.g.[3]- [27] and the references therein.
In this paper we study the existence of a global random attractor for stochastic strongly damped wave equations with multiplicative noise d d . The coefficient c of the noise term needs to be suitable small, which is different from that in stochastic strongly damped wave equations with additive noise, this is because the multiplicative noise depends on the state variable u but the additive noise term is independent of u .This paper is organized as follows.In the next section, we recall some basic concepts and properties for general random dynamical systems.In Section 3, we provide some basic settings about Equation (1.1) and show that it generates a random dynamical system in proper function space.Section 4 is devoted to proving the existence of a unique random attractor of the random dynamical system.

Preliminaries
In this section, we collect some basic knowledge about general random dynamical systems (see [28] [29] for details).

Let ( )
, X X ⋅ be a separable Hilbert space with Borel σ -algebra ( ) be a probability space as in Section 1. Define ( ) is an ergodic metric dynamical system [28] [29].
In the following, a property holds for  -a.e.ω ∈ Ω means that there is 0 Ω ∈ Ω with ( )

{ }
A ω is said to be a random attractor if it is a random attracting set and for  -a.e.ω ∈ Ω and all 0 t ≥ .Theorem 2.3 (See [29]).Let ϕ be a continuous random dynamical system on X over

{ }
A ω is the unique random attractor of ϕ .

Stochastic Strongly Damped Wave Equation
In this section, we outline the basic setting of (1.1)-(1.2) and show that it generates a random dynamical system.Define an unbounded operator Clearly, A is a self-adjoint, positive linear operator with the eigenvalues { } i i λ ∈ : ( ) for , , where ⋅ denotes the usual norm in ( )

2
L U and Τ stands for the transposition.It is convenient to reduce (1.1) to an evolution equation of the first order in time For our purpose, it is convenient to convert the problems (1.1)-(1.2) into a deterministic system with a random parameter, and then show that it generates a random dynamical system.
We now introduce an Ornstein-Uhlenbeck process given by the Brownian motion.Put which is called Ornstein-Uhlenbeck process and solves the Itô equation ( ) From [30] [31], it is known that the random variable ( ) z ω is tempered, and there is a t θ -invariant set 0 Ω ⊂ Ω of full  measure such that ( ) t z θ ω is continuous in t for every 0 ω ∈ Ω .
generates a continuous random dynamical system over also generates a continuous random dynamical system associated with the problem (3.2) on E .Note that the two random dynamical systems Ψ and Φ are equivalent.By transformation (3.11), it is easy to see that Ψ has a random attractor provided Φ possesses a random attractor.Thus, we only need to consider the random dynamical system Φ .

Random Attractor
In this section, we study the existence of a random attractor.Throughout this section we assume that ( ) is the collection of all tempered random subsets of E and ( ) ( ) For our purpose, we introduce a new norm For ( ) where , ⋅ ⋅ denotes the inner product on ( ) 2 L U .By the Poincaré inequality ( ) 3) is then positive definite.Now, we present a property of the operator M ε in E that plays an important role in this article.Lemma 4.1 Let ( ) ( ) . There exists a small positive constant ( ) 2 The proof of Lemma 4.1 is similar to that of Lemma 1 in [24].We hence omit it here.
Proof.Take the inner product , E ⋅ ⋅ of problem (3.9) with φ .By the Cauchy-Schwartz inequality and the Young inequality, we find that ( ) ( ) , , where U is the volume of the set U .By using the Poincaré inequality (4.4), we have that By all the above inequalities and Lemma 4.1, we have By the Gronwall lemma, we have that, for all By replacing ω by t θ ω − , we get from problem (4.8) that, By inequality (4.1), it is easy to see that It then follows from inequality (4.10), Lemma 3.1, ( ) ( ) By Lemma 3.1, inequality (4.10) and ( ) Then, for any tempered random set such that for any ( ) ( ) , , .
So, the proof is completed.


We now construct a random compact attracting set for RDS Φ .For this purpose, we decompose the solution φ of Equation (3.9) with the initial value ( ) ( ) ( ) ( ) ( ) ( ) where a φ and b φ satisfy Equations (4.15), (4.16).Proof.We first take the inner product , E ⋅ ⋅ of Equation (4.15) with a φ .By Lemma 4. Aφ .From the positivity of the operator A , we easily obtain By the Cauchy-Schwartz inequality and the Young inequality, we find that ( ) ( ) ( ) By using inequality (4.4), we have that Combining all the above inequalities and inequality (4.21), we have Using the Gronwall lemma, for all 0 t ≥ , we get , and for some constant 0 κ > , we have for  -a.e.ω ∈ Ω , ( : .