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We study the asymptotic behavior of solutions to the stochastic strongly damped wave equation with additive noise defined on unbounded domains. We first prove the uniform estimates of solutions, and then establish the existence of a random attractor.

Let

the Borel

Thus,

Consider the following stochastic strongly damped wave equation with additive noise defined in the entire space

with the initial value conditions

where

function satisfying certain dissipative and growth conditions, and

valued Wiener processes on

Many works have been done regarding the dynamics of a variety of systems related to Equation (1). For example, the asymptotical behavior of solutions for deterministic strongly damped wave equation has been studied by many authors (see [

In general, the existence of global random attractor depends on some kind compactness (see, e.g., [

Throughout this paper, we use

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

In the following, a property holds for

Definition 1 A continuous random dynamical system on X over

such that the following properties hold

・

・

・

Definition 2 (See [

・ A set-valued mapping

・ A random set

where

Let

・ A random set

・ A random set

where

・

・ A random compact set

Theorem 1 (See [

Moreover,

In this subsection, we outline some basic settings about (1)-(2) and show that it generates a random dynamical system.

Let

with the initial value conditions

where

Let

where

For our purpose, it is convenient to convert the problem (3)-(4) (or (1)-(2)) into a deterministic system with a random parameter, and then show that it generates a random dynamical system.

Let

Its unique stationary solution is given by

Note that the random variable

where

Then it follows from the above, for

Put

Now, let

with the initial value conditions

where

Let

where

By a standard method as in [

generates a continuous random dynamical system, where

Then, the transformation

also generates a random dynamical system associated with (3)-(4). Note that the two random dynamical systems are equivalent. By (13), it is easy to check that

In this subsection, we derive uniform estimates on the solutions of the stochastic strongly damped wave Equations (3)-(4) defined on

We assume that

Set

where

We define a new norm

for

The next lemma shows that

Lemma 1 Assume that (F1)-(F4),

dom ball

sorbing set for

Proof. Taking the inner product of the second equation of (9) with

By the first equation of (9), we have

Then substituting the above

From conditions (F1)-(F3) we get

Using the Cauchy-Schwartz inequality and the Young inequality, we have

By (19)-(24), it follows from (17) that

Recalling the new norm

Using the Gronwall lemma, we have

Substituting

By (5), we get

By assumption,

Note that

By (F3), we have that

Combining (28), (30), (31) and (32), there is a

where

To prove asymptotic compactness of the random dynamical system

Given

Choose a smooth function

and there exist constants

Lemma 2 Assume that (F1)-(F4),

lution

Proof. We first consider the random Equations (9)-(10). Then taking the inner product of the second equation

of (9) with

Substituting

By using conditions (F1), (F2) and (F3), we find

By the Cauchy-Schwartz inequality and the Young inequality, we obtain

Then it follows from (37)-(42) that

Letting

then, by (14) we have from (43) that

By using the Gronwall lemma, we get that

By replacing

By using (F3), there exists

In what follows, we estimate the terms on the right-hand side of (47). By (5),

Since

Note that

Next, we estimate the forth term on the right-hand side of (47). Using (F3), replacing t by s and then

it then follows that

Since

Letting

which implies

Then we complete the proof.

Let

Multiplying (9) by

Considering the eigenvalue problem

The problem has a family of eigenfunctions

such that

Lemma 3 Assume that (F1)-(F4),

Proof. Let

Then applying

Substituting

Using conditions (F1) and (F4), we have

it then follows that

By using the Cauchy-Schwartz inequality and the Young inequality, we have

From (63)-(73) we can obtain that

Since

Using the Gronwall lemma, we have

By substituting

We next estimate each term on the right-hand side of (77). Since

Since

term on the right-hand side of (77) satisfies

Next, we estimate the third term on the right-hand side of (77). By (6), (18) and (33),

which implies that there exists

Let

which completes the proof.

In this subsection, we prove the existence of a global random attractor for the random dynamical system generated by (9)-(10).

Theorem 2 Assume that (F1)-(F4),

Proof. Notice that the random dynamical system

Next, we will prove that the random dynamical system

Let

is a bounded in

By Lemma 2, we have that there are

In addition, it follows from Lemma 3 that there exist

Then, by (57) and (83),

implies that

dynamical system

Then, by Theorem 1, the random dynamical system

In the present article, we have discussed the existence of a random attractor to the stochastic strongly damped wave equation with additive noise defined on unbounded domains. It is also interesting to consider the the same

problem for stochastic strongly damped wave equation with multiplicative noise

coefficient

noise

in the future.

We thank the editor and the referee for their comments. 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); Zhejiang Natural Science Foundation under Grant No. LY14A010012 and Zhejiang Normal University Foundation under Grant No. ZC304014012. This support is greatly appreciated.