The Random Exponential Attractor For Stochastic Kirchhoff Equation And P-Laplacian Equation

This paper studies the existence of random exponential attractor for two kinds of non-autonomous stochastic partial differential equations with widespread applica-tion background.Random attractor may be infinite dimensional and sometimes attracts orbits at a relatively slow rate so that it takes a long time to reach it,but random expo-nential attractor has these characters of finite fractal dimension,attracting exponentially any trajectory and positively compact invariant.Thus random exponential attractor be-comes one of important ways of the infinite dimensional dynamical systems.The two kinds of partial differential equation that this paper considers are defined on unbounded domain(R~3,Q).Notice that the Sobolev embedding theorem is invalid on unbounded domain.This introduces a major obstacle for proving the existence of random attractors.This article utilize the method of tail estimation to overcome the obstacle.Finally,we prove the existence of random attractor for two kinds of partial differential equation.The framework of this paper as follows.In the first chapter,we mainly introduce the status of random dynamical system at home and abroad.Meanwhile,the notion of continuous cocycle and random exponen-tial attractor,classical inequality and Sobolev embedding theorem are given.In the second chapter,we study the existence of random exponential attractor for non-autonomous Kirchhoff equation with damping and multiplicative white noise.Firstly,by the existence of absorbing set,estimation on tail of solutions,we prove the existence of random attractor.Secondly,we apply the method of estimating the bound-edness of fractal dimension and Lipschitz continuous to prove the existence of random exponential attractor for the Kirchhoff equation.In the third chapter,we prove the existence of random exponential attractor for non-autonomous p-Laplacian equation with multiplicative white noise.Firstly,we use the way of estimation on tail of solutions to prove the existence of random attractor.Then,we utilize decomposition of solutions to get the Lipschitz continuous of the sys-tem.Finally,by the way of estimating the boundedness of fractal dimension for random attractor and the expectations of some random variables,we prove the existence of ran-dom exponential attractor for the p-Laplacian equation.In the fourth chapter,we make some conclusions about this article and put forward new problem remaining to be researched.