Existence of Random Attractor Family for a Class of Nonlinear Higher-Order Kirchhoff Equations

The existence of random attractor family for a class of nonlinear nonlocal higher-order Kirchhoff partial differential equations with additive white noise is studied. The weak solution of the equation is established by the Ornstein-Uhlenbeck process to deal with the random term, and a bounded random absorption set is obtained. And then, the existence of the random attractor family is proved by the isomorphism mapping method.


Introduction
In this paper, we consider the following of nonlinear strongly damped stochastic Kirchhoff equations with additive white noise: where 1 m > is a positive integer, β is a normal number, Ω is a bounded region with smooth boundary in R n , M is a general real-valued function, Int. J. Modern Nonlinear Theory and Application term. The assumptions about M and g will be given later.
Xintao Li and Lu Xu [1] have studied the following stochastic delay discrete wave equation The existence of random attractors for this equation is proved by means of tail-cutting technique and energy estimation under appropriate dissipative conditions.
Ailing Ban [2] have considered the following of stochastic wave equations β > is the damping coefficient, and 0 K > is the dissipation coefficient. In this paper, they mainly discuss the asymptotic behavior of strongly damped stochastic wave equation with critical growth index. By using the weighted norm, they prove that for any positive strong damping coefficient and dissipation coefficient, there is a compact attractor for the stochastic dy They mainly use the Ornstein-Uhlenbech process to deal with the stochastic term of Equation (11), thus obtain the global well-posedness of the solution, and then prove the existence of the global random attractor.
As we all know, attractors have absorptivity and invariance, and have a clear description of the long-term behavior and the asymptotic stability of the solution of the equation. Because the long-term behavior of the system develops within the overall attractor, and then on this compact set, through the study of the overall behavior characteristics of the system, we can find the most common rules of the system and the basic information of future development. In real life, the evolution of many problems will be disturbed by some uncertain factors. At this time, the deterministic dynamic system can no longer describe these problems.
Therefore, it is necessary to study the attractors of stochastic equations with additive noise terms.  [12].
The structure of this paper is as follows: in Section 2, some basic assumptions and knowledge of dynamical system required in this paper are introduced; in Section 3, the existence of random attractor family subfamilies is proved by using the isomorphism mapping method.

Basic Hypothesis and Elementary Knowledge
In this section, some symbols, definitions and assumptions about Kirchhoff type stress term ( ) M s and nonlinear nonlocal source term ( ) , t g x u are given. In addition, some basic definitions of stochastic dynamical systems are also introduced.
For narrative convenience, we introduce the following symbols: It is assumed that the Kirchhoff type stress term The following will introduce some basic knowledge about random attractor.
is an orbiting metric dynamical system.

Let ( )
, X X ⋅ be a complete separable metric space and ( ) B X be a Borel σ -algebra on.
Definition 1 (Following as [12] be a metric dynamical system, suppose that the mapping -measurable mapping and satisfies the following properties: 1) The mapping ( ) ( ) Then S is a continuous stochastic dynamical system on Definition 2 (Following as [12]) It is said that random set ( ) Definition 3 (Following as [12]) Let ( ) D ω be the set of all random sets on X, and random set Definition 4 (Following as [12]) Random set ( ) A ω is called the random attractor of continuous stochastic dynamical system ( ) S t on X , if random set

( )
A ω attracts all sets on .

Existence of Random Attractor Family
In this section, we mainly consider the existence of random attractor family of , then the question (14) can be simplified to , .
, Then the question (14) may read as follows: Proof: Let ψ is a solution of the problem (16), taking inner product of two Int. J. Modern Nonlinear Theory and Application sides of the Equation (15) is obtained by using From Lemma 1, we have ( ) According to Holder inequality, Young inequality and Poincare inequality, we from assumption (A2), we have B ω is a random absorb set of ( ) , t ϕ ω , and Thus, the whole proof is complete.
It is shown below that there exists a compact suction collection for stochastic dynamical system ( )