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In this paper, we consider the stochastic higher-order Kirchhoff-type equation with nonlinear strongly dissipation and white noise. We first deal with random term by using Ornstein-Uhlenbeck process and establish the wellness of the solution, then the existence of global random attractor are proved.

In this paper, we consider the following stochastic strongly damped higher- order nonlinear Kirchhoff-type equation with white noise:

with the Dirichlet boundary condition

and the initial value conditions

where

1)

2)

3)

4)

where

As well as we known, the study of stochastic dynamical is more and more widely the attention of scholars, and the study of random attractor has become an important goal. In a sense, the random attractor is popularized for classic determine dynamical system of the global attractor. Global attractor of Kirchhoff- type equations have been investigated by many authors, see, e.g., [

In this work, we deal with random term by using Ornstein-Uhlenbeck process, the key is to handle the nonlinear terms and strongly damped

This paper is organized as follows: In Section 2, we recall many basic concepts related to a random attractor for genneral random dynamical system. In Section 3, we introduce O-U process and deal with random term. In Section 4, we prove the existence of random attractor of the random dynamical system.

In this section, we collect some basic knowledge about general random dy- namical system ( [

Let

Definition 2.1. (see [

1)

2)

3)

Definition 2.2. (see [

1) A set-valued mapping

2) A random set

where

Let Y be the set of all random tempered sets in X.

3) A random set

4) A random set

where

5)

6) A random compact set

Theorem 2.1. ( [

Moreover,

Let

Let

O-U process is given by Wiener process on the metric system

Let

equation:

1) For

2) Random variable

3) Exist temper set

4)

5)

For convenience, we rewrite the Question (1.1)-(1.3):

Let

where

Let

where

In [

Theorem 3.2.1. Consider (3.2.3). For each

and

For

generates a random dynamical system.

Define two isomorphic mapping:

And inverse isomorphic mapping:

Then the mapping

Notice that all of the above random dynamical system

First, we prove the random dynamical system

Lemma 4.1. (Lemma 3.1 of [

where

Lemma 4.2. Let

Proof. Let

where

According to (4.1) and (4.4)-(4.10), we have

where

According to Gronwall inequation,

Because

Substituting

where

Because

then

so let

Next, we will prove the random dynamical system

Lemma 4.3. For

Then

and exist a temper random radius

Proof. Let

Taking inner product (4.21) with

according to Lemma 4.1 and Gronwall inequality, we have

substituting

So, (4.19) is hold. Taking inner product (4.22) with

according to Lemma 4.1, Lemma 4.2, (4.24) and Young inequality, we have

where

According to (4.14) and (4.16), then

Let

Then

hence, we set

Lemma 4.4. (3.2.2) the identified stochastic dynamical system

Proof. Let

set in E, for arbitrarily temper random set

+

Theorem 4.1. The random dynamical system

in which

Lin, G.G., Chen, L. and Wang, W. (2017) Random Attractor of the Stochastic Strongly Damped for the Higher-Order Nonlinear Kirchhoff-Type Equation. International Journal of Modern Nonlinear Theory and Application, 6, 59-69. http://dx.doi.org/10.4236/ijmnta.2017.62005