Random Attractors For Stochastic Strongly Damped Wave Equations

In this dissertation we mainly study the existence,upper boundedness of fractal dimension of random attractors for autonomous and non-autonomous stochastic strongly damped wave equations.This dissertation is divided into three chapters.In chapter 1,we first introduce the background and current situation of the research.Then we present some basic concepts,lemmas and theorems related with attractors for random dynamical system and some inequalities which will be applied in this dissertation.At last,we introduce the main work obtained in this dissertation of wave equation.In chapter 2,we mainly consider the upper bound estimation of the fractal dimension of random attractors for stochastic strongly damped wave equations with additive and multiplicative white noises.Firstly,we transfer these stochastic equations with white noises into random equations with random parameters but without noise terms by the Ornstein-Uhlenbeck process.Secondly,we divide the difference between the two solutions of random equations into a sum of two parts,and estimate the norm of these two parts and the boundedness of expectation of some random variables,respectively.Finally,we obtain the upper bound estimation formula of the fractal dimension of random attractors for the considered equations.In chapter 3,we mainly consider the existence of random attractor for non-autonomous stochastic strongly damped wave equations with multiplicative white noises.Firstly,we transfer these stochastic equations with white noises into random equations with ran-dom parameters but without noise terms by the Ornstein-Uhlenbeck process;Secondly,through the estimation of the solution of equation,we get a bounded random absoring set;Finally,we obtain asymptotic compactness of the RDS by the sobolev embedding theorem,and prove the existence of random attractor.