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

Guoguang Lin,Liping Guan

doi:10.5539/jmr.v11n3p23

Abstract

A class of stochastic dynamical systems with strong damped stochastic higher order Kirchhoff equation solutions with white noise is studied. Firstly, the equation is transformed into a stochastic equation with random variables as parameters and without noise by using Ornstein-Uhlenbeck process. Secondly, the bounded stochastic absorption set is obtained by estimating the solution of the equation. Finally, the stochastic dynamical system is obtained by using the isomorphic mapping method and the compact embedding theorem. It is progressively compact, thus proving the existence of random attractors.

Highlights

In this paper, the stochastic higher-order Kirchhoff equation with strong damping and additive noise is studied.utt M ( Dmu 2 )( )m u ( )m ut g(u) q(x)W, (1.1)u x,t 0, i u 0, i 1,2, m 1, x D, t [0, ), vi (1.2)u x,0 u0 x, ut (x,0) u1(x). (1.3)Where m 1, g(u) is a second-order non-linear source term, M is a real-valued function, 0. u u(x,t) is a real-valued function on D [0, ), D is a bounded open set with smooth boundary on Rn (n N) . qdW describes an additive white noise
The equation is transformed into a stochastic equation with random variables as parameters and without noise by using Ornstein-Uhlenbeck process
The stochastic dynamical system is obtained by using the isomorphic mapping method and the compact embedding theorem

Summary

Introduction

The stochastic higher-order Kirchhoff equation with strong damping and additive noise is studied. By using Ornstein-Uhlenbeck process and isomorphic mapping method, the existence and uniqueness of solutions and the existence of random attractors for stochastic Kirchhoff equation with strong damping are obtained. The existence of random attractors for Nonautonomous 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.

Preliminaries

Existence of Random Attractor Family