A Family of the Random Attractors for a Class of Generalized Kirchhoff-Type Equations

In this paper, we studied the existence of a family of the random attractor for a class of generalized Kirchhoff-type equations with a strong dissipation term. Firstly, according to Ornstein-Uhlenbeck process, we transformed the equation into a stochastic equation with random variables and multiplicative white noise. Secondly, we proved the existence of a bounded random absorbing set. Finally, by using the isomorphic mapping method and the compact embedding theorem, we get the stochastic dynamical system with a family of random attractors.


Introduction
In recent years, the global attractor, exponential attractor, inertial manifold, and approximate inertial manifold of the Kirchhoff equation in infinite dimensional dynamical systems have been extensively studied. With further in-depth research, people have found that many real-life problems will be interfered with by all external uncertain factors to varying degrees, and a deterministic dynamic system cannot be used to describe this type of problem. At this time, we introduce a random attractor with multiplicative white noise. The random attractor is a measurable, compact and invariant random set that attracts all solution orbits.
deterministic dynamic system, so the random attractor has more practical and deeper properties. The random attractors can be used to study fluid mechanics, finance and other fields; they are the supplement of deterministic dynamical systems. Therefore, many scholars have done a lot of research on random attractors of nonlinear partial differential equations with white noise, and have obtained a series of research results, including stochastic parabolic equations, generalized Ginzburg-Landau equations, dissipative KdV equation, stochastic reaction-diffusion equations, stochastic Sine-Gordon equations, stochastic Boussinesq equations, stochastic Kirchhoff equations and other stochastic evolution equations have corresponding study about random attractors, more significant research can refer to [1]- [10]. Guoguang Lin, Ling Chen, Wei Wang [11] studied the stochastic strongly damped higher-order nonlinear Kirchhoff-type equation with white noise: They proved the existence of a random attractor of the random dynamical system.
Guigui Xu and Libo Wang [12] studied the large-time behavior of the following initial boundary value problem for the stochastic strongly damped wave equation with white noise in a bounded domain R ⊂  with smooth boundary:  , and , α β are positive constants, . W is a scalar Gaussian white noise, that is, ( ) W t is a two-sided wiener process.

The functions
: satisfies the following assumptions:  is not identically equal to zero; 2) The nonlinear term f satisfies where 0 1 2 , , C C C are positive constants.
Guoguang Lin and Zhuoxi Li [13] studied the random attractor family of solutions to the strongly damped stochastic Kirchhoff equation with white noise: They get the temper random compact sets of random attractor family.
On the basis of reference [13], the stress term 2 m D u is extended to p m p D u , Journal of Applied Mathematics and Physics this paper studied the long-time dynamic behavior of a class of generalized Kirchhoff equation. According to preliminary knowledge and reasonable assumption for Kirchhoff stress term and nonlinear source term, we proved the existence of random absorbing set in stochastic dynamical system; furthermore, a family of the random attractor is obtained.
In this paper, we study the existence of a family of the random attractors for a class of generalized Kirchhoff-type equation with damping term: bounded domain with a smooth boundary ∂Ω , d q W denotes an additive white noise.

( )
W t is a one-dimensional bilateral Wiener process on probability , F is a Borel σ -algebra generated by compact open topology on Ω , p is a probability measure, the assumption of ( ) is Lipschitz continuous; (A2) There existence constant 0 g l > , such that µ µ µ are constant, 1 λ is the first eigenvalue of −∆ with homogeneous Dirichlet boundary conditions on Ω .

Preliminaries
For convenience, define the following spaces and notations: Here are some basic knowledge of stochastic dynamic systems required: is a probabilistic space and define a family of measures-preserving and ergotic transformations of { } is an ergodic metric dynamical system.
endowed with compact-open topology, P is the corresponding Wiener measure, and F is the Borel σ -algebra on Ω . The space is called the metric dynamical system on the probability space ( ) A w is called a random attractor on X for continuous stochastic dynamical system . a e w P ∈ Ω , we have the limit formula has a unique global attractor, i.e., The Ornstein-Uhlenbeck process is given as following: From the above we can know that the Ornstein-Uhlenbeck process on 2) The random variable ( ) z w is tempered; 3) There exist a slowly increasing set ( ) 0 r w > , such that

The Existence for a Family of the Random Attractor
In this section, our objection is to prove the existence of random attractors for the initial boundary value problem (1.1)-(1.3).
At first, we define the inner product and norms on k E as follows: , .
is a stochastic process, then Equation (3.1) can be written as so there's a non-negative real number Proof. Taking the inner product of the second equation of (3.1) with v in ( ) 2 L Ω , we find that  , , The following estimation can be obtained from hypothesis (A1) By using the weighted Young's inequality, we obtain Then ( ) ( ) So, there are constants Proof. Because of ( )