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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.

Consider the following stochastic strongly damped wave equation with multiplicative noise:

with the homogeneous Dirichlet boundary condition

and the initial value conditions

where

where

the Borel

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. [

In this paper we study the existence of a global random attractor for stochastic strongly damped wave equations with multiplicative noise

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

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.

In this section, we collect some basic knowledge about general random dynamical systems (see [

Let

then

In the following, a property holds for

Definition 2.1 A continuous random dynamical system on X over

which is

1)

2)

3)

Definition 2.2 (See [

1) A set-valued mapping

is called a random closed (compact) set. A random set

2) A random set

3) A random set

4) A random set

where

5) A random compact set

Theorem 2.3 (See [

Moreover,

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,

It is well known that

Let

where

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 [

Lemma 3.1 (See [

To show that problem (3.2) generates a random dynamical system, we let

which

which has the following vector form

where

We will consider Equation (3.8) or (3.9) for

By the classical theory concerning the existence and uniqueness of the solutions [

generates a continuous random dynamical system over

Introduce the homeomorphism

also generates a continuous random dynamical system associated with the problem (3.2) on

Note that the two random dynamical systems

In this section, we study the existence of a random attractor. Throughout this section we assume that

For our purpose, we introduce a new norm

for

where

Equation (4.3) is then positive definite.

Now, we present a property of the operator

Lemma 4.1 Let

The proof of Lemma 4.1 is similar to that of Lemma 1 in [

Lemma 4.2 Assume that

Proof. Take the inner product

where

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

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

We choose

Then, for any tempered random set

So, the proof is completed.

We now construct a random compact attracting set for RDS

Lemma 4.3 Assume that

and there exist a tempered random variable

where

Proof. We first take the inner product

Then by

Thus, the first assertion is valid.

Next, we take the inner product

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

Replacing

By Lemma 3.1, inequality (4.10) and

We can choose

then the second assertion is valid.

By Lemma 4.2 and Lemma 4.3, for any

where

Then, by the compact embedding of

Note that

Then by Lemma 4.3 and inequality (4.27), we have for

which implies that

Theorem 4.4 Assume that

in which

The authors are supported by National Natural Science Foundation of China (Nos. 11326114, 11401244, 11071165 and 11471290); Natural Science Research Project of Ordinary Universities in Jiangsu Province (No. 14KJB110003); the Foundation of Zhejiang Normal University (No. ZC304011068).

ZhaojuanWang,ShengfanZhou, (2015) Asymptotic Behavior of Stochastic Strongly Damped Wave Equation with Multiplicative Noise. International Journal of Modern Nonlinear Theory and Application,04,204-214. doi: 10.4236/ijmnta.2015.43015