Random Attractor Family for the Kirchhoff Equation of Higher Order with White Noise

The existence of random attractor family for a class of nonlinear high-order Kirchhoff equation stochastic dynamical systems with white noise is studied. The Ornstein-Uhlenbeck process and the weak solution of the equation are used to deal with the stochastic terms. The equation is transformed into a general stochastic equation. The bounded stochastic absorption set is obtained by estimating the solution of the equation and the existence of the random attractor family is obtained by isomorphic mapping method. Temper random compact sets of random attractor family are obtained.


Introduction
In this paper, we study the random attractor family of solutions to the strongly damped stochastic Kirchhoff equation with white noise: , with the Dirichlet boundary condition ( ) , 0, 0, 1, 2, , 1, , 0, and the initial value conditions , 0 , , 0 , , where 1 m > is a positive integer; 0 β > is a constant; Ω is a bounded re- gion with smooth boundary in n R .∆ is the Laplacian with respect to the va- G. G. Lin ,  g x u is a non-linear and non-local source term.W is derivative of a one-dimensional two-valued Wiener process ( ) W t and ( ) q x W  formally describes white noise.B.L. Guo and X.K. Pu described in detail the related concepts and theories of infinite dimensional stochastic dynamical systems, and discussed in detail the existence and uniqueness, attractor and inertial manifold of some nonlinear evolution equations and wave equation solutions in [1] [2].D.H. Cai and X.M. Fan [3], considered the dissipative KDV equation with multiplicative noise.Yin et al. [4] have mainly studied the dissipative Hamiltonian amplitude modulated wave instability equation with multiplicative white noise.
( ) Stochastic dynamic system has compact random attractors in space Xu et al. [5] studied the non-autonomous stochastic wave equation with dispersion and dissipation terms.
The existence of random attractors for non-autonomous stochastic wave equations with product white noise is obtained by using the uniform estimation of solutions and the technique of decomposing solutions in a region.
Lin et al. [6] studied the existence of stochastic attractors for higher order nonlinear strongly damped Kirchhoff equation. ( The O-U process is mainly used to deal with the stochastic terms, and the existence of stochastic attractors is obtained.
Qin et al. [7] studied random attractors for the Kirchhoff-type suspension bridge equations with Strong Damping and white noises.
Kirchhoff stress term ( ) ∆ and dissipation term bu + are treated.
It is assumed that the non-linear term

( )
f u satisfies the growth and dissipation conditions.
For more relevant studies, it can be referred to references in [8]- [13].
On the basis of some random attractors of Kirchhoff equation with white noise studied by predecessors, the existence and uniqueness of solutions of stochastic higher-order Kirchhoff equation with strong damping of white noise, nonlinear and non-local source terms and the existence of attractors of stochastic Kirchhoff equation are discussed.This paper is organized as follows.In Section 2, some basic assumptions and basic concepts related to random attractor for general random dynamical system are presented.Section 3 deals with random term and proof the existence of random attractor family by using the isomorphism mapping method.

Preliminaries
In this section, some symbols are made and assumption Kirchhoff Stress term ( ) M s satisfying condition (a) and Nonlinear term ( ) , g x u satisfies condition (b).In addition, some basic definitions of stochastic dynamical systems are also introduced.
For narrative convenience, we introduce the following symbols: is an ergodic metric dynamical system.

Let ( )
, X X ⋅ be a complete separable metric space and ( ) -measurable mapping and satisfies the following properties: 1) The mapping ( ) ( ) w id S t s w S t w S s w Then S is a continuous stochastic dynamical system on lim inf e 0 A w called the random attractor of continuous stochastic dynamical systems ( ) S t on X, if random set ( ) A w satisfies the following conditions: 1) ( ) A w is a random compact set; 2) ( ) 3) ( ) A w attracts all the set on ( ) D w , that is, for any ( ) ( ) . .P a e w − ∈ Ω , with the following limit: where ( )  ( ) The Ornstein-Uhlenbeck process [7] is given as following.

The Existence of Random Attractor Family
In this section, we consider the existence of random attractor family.To deal with the random term need to transform the problem (1.1) -( 1.3) into a general stochastic problem.It is proved that there exists a bounded stochastic absorption set for stochastic dynamical systems.The stochastic dynamical system exists stochastic attractor family and a slowly increasing stochastic compact set.

Let
, , , then the question (3.2) can be written as: , , where ( ) Proof: For any ( ) , y y y = , according to hypothesis (a), we have

Ly y y y y M A u
where ( ) Let φ be a solution of the problem (3.2), then there exists a bounded random compact set ( ) ( ) , so that for any random set ( ) ( ) , there exists a random variable Proof: Let ϕ be a solution of the problem (3.3), by taking the inner product of two sides of the Equation (3.3) is obtained by using From Lemma 1, we have ( ) , .
According to the inner product defined on k E .
According to Holder inequality, Young inequality and Poincare inequality, we have According to hypothesis (b), we have Combining (3.8)-(3.13)yields, we have .
, , 3), then according to the Equation (3.24) and (3.25), we can see that 1 2 , ϕ ϕ meet separately ( ) By taking the inner product of equation within m E , we have ( )

4 )
By transforming the equation into a stochastic KDV-type equation without white noise, the existence of stochastic attractors for dynamic systems determined by the original equation is proved by discussing the dynamic absorptivity and asymptotic property determined by the new equation.
is a closed set on Hilbert space X.
the problem (3.1) can be simplified to: