Random Attractor of the Stochastic Strongly Damped for the Higher-Order Nonlinear Kirchhoff-Type Equation

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.


Introduction
In this paper, we consider the following stochastic strongly damped higherorder nonlinear Kirchhoff-type equation with white noise: , with the Dirichlet boundary condition ( ) , 0, 0, 1, 2, , 1, , and the initial value conditions ,0 , ,0 , where Ω is a bounded domain of n R , with a smooth boundary ∂Ω , ∆ is the Laplacian with respect to the variable x ∈ Ω , ( ) is a real function of x ∈ Ω and 0 t ≥ , φ is the damping coefficient, f is a given external force, v is the outer norm vector, ( ) g u is a nonlinear forcing, their respectively satis- fies the following conditions: 1) ( ) ( ) ( ) s m φ ≥ where 0 1 2 0 , , , c c c m are positive constants.
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 Kirchhofftype equations have been investigated by many authors, see, e.g., [1] [2] [3] [4], however, the existence random attractor has also been studied by many authors, in [5], Zhaojuan Wang, Shengfan Zhou and Anhui Gu, they study the asymptotic dynamics for a stochastic damped wave equation with multiplicative noise defined on unbounded domains, and investigate the existence of a random attractor, they overcome the difficulty of lacking the compactness of Sobolev embedding in unbounded domains by the energy equation.In [6], Guigui Xu, Libo Wang and Guoguang Lin study the long time behavior of solution to the stochastic strongly damped wave equation with white noise, in this paper, they use the method introduced in [7], so that they needn't divide the equation into two parts.In [8], Zhaojuan Wang, Shengfan Zhou and Anhui Gu study the asymptotic dynamics of the stochastic strongly damped wave equation with homogeneous Neuman boundary condition, and prove the existence of a random attractor.The other long time behavior of solution of evolution equations, we can see [9]- [19].
In this work, we deal with random term by using Ornstein-Uhlenbeck process, the key is to handle the nonlinear terms and strongly damped ( ) is also difficult to be conducted.So far as we know, there were no result on random attractor for the stochastic higher-order Kirchhoff-type equation with nonlinear strongly dissipation and white noise.It is therefore important to investigate the existence of random attractor on (1.1)- (1.3).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.

Let ( )
, .X X be a separable Hilbert space with Borel σ-algebra ( )  Let Y be the set of all random tempered sets in X.
where H d is the Hausdorff semi-distance given by ( ) 5) ϕ is said to be asymptotically compact in X if for has a convergent subsequence in X whenever n t → +∞ , and ( )

6) A random compact set ( )
A ω is said to be a random attractor if it is a random attracting set and ∈ Ω and all 0 t ≥ .Theorem 2.1.( [10]) Let ϕ be a continuous random dynamical system with state space X over If there is a closed random absorbing set ( ) B ω of ϕ and ϕ is asymptotically compact in X, then ( ) A ω is the unique random attractor of ϕ .

O-U Process and Stochastic Dynamical System
Let ( ) Ω , and define a weighted inner product and norm in E ( ) ( ) ( ) , , ,

O-U Process
O-U process is given by Wiener process on the metric system we can see ( [11] [12] [13]).
Let ( ) ( ) And there is a probability measure P, t θ -in- variant set Ω ⊂ Ω 0 ; so that stochastic process ( ) ( ) meet the following properties: 1) For

Stochastic Dynamical System
For convenience, we rewrite the Question (1.1)-(1.3): , and µ ε = ( ε defined in [20]), then (3.2.1) has the following simple matrix form , then (3.2.1) can be rewritten as the equivalent system: In [14] [15] they have proven that the operator L of ( ∈ , there exists a unique function γ such that satisfies the integral equation generates a random dynamical system. Define two isomorphic mapping:  Notice that all of the above random dynamical system ( )

The Existence of Random Attractor
First, we prove the random dynamical system ( ) , S t ω exists a bounded random absorb set, hence we let ( ) D E be all temper subsets in E.
where 1 2 , k k are determined in [20], λ is first eigenvalues of (1.1).Lemma 4.2.Let ψ is a solve of (3.2.2), then there is a bounded random com- pact set , such that for arbitrarily random set ( ) ( ) Proof.Let γ is a solve of (3.2.3), applying the inner product of the equation According to (4.1) and (4.4)-(4.10),we have where According to Gronwall inequation, . .
Because ( ) t z θ ω is tempered, and ( ) t z θ ω is continuous about t, according to [21], we can get a temper random variables , .
absorb set, and because of ( Next, we will prove the random dynamical system ( )  ( ) ( ) and exist a temper random radius , .
Due to Gronwall inequality, and substituting ω by

=
generates a random dynamical system associated with (3.2.2).

K
ω is a compact set in E, for arbitrarily temper random set ( ) B ω , for ( ) 3.2.3) is the infinitesimal generation operator of 0C semigroup e Lt in Hilbert space E, is continuous in t and globally Lipschitz continuous in γ for each ω ∈ Ω .By the classical theory concerning the existence and uniqueness of the solutions[14] [16][17], so we have the following theorem.