Asymptotic Behavior Of Some Nonlinear Stochastic Differential Equations

At present,famous domestic scholars and foreign scholars have done a lot of research on random attractor theory.In the results of the current study,the noise of stochastic dynamical systems is usually additive white noise or multiplicative white noise,the non-autonomous equations with additive white noise or multiplicative white noise are studied by transforming them into deterministic equations.However,there is no method to transform non-autonomous equations with nonlinear white noise into deterministic equations.Inspired by.(2(69)2)~?2019 paper in the Jour-nal of Dynamics and Differential Equations,in my dissertation,I study the non-autonomous wave equations with nonlinear white noise and the non-autonomous Benjamin-Bona-Mahony equations(BBM)with nonlinear white noise by using the theory of weak pullback mean random attractors.In fact,these dynamical systems are mean-valued stochastic dynamical systems whose solutions are defined in Bochner space.Firstly,the existence and uniqueness of the solutions of the above equation-s are proved by using uniform estimates,then the existence and uniqueness of the weakly compact absorbing sets is proved by means estimation,finally,it is proved that the mean random dynamical system generated by the operator has a unique weak pullback mean random attractor in Bochner space.In Chapter 1,I introduce the background and present situation related weak pullback mean random attractors,and give the related concepts and common in-equalities.In Chapter 2,I study the existence and uniqueness related weak pullback meanrandom attractors for mean stochastic dynamical systems generated by non-autonomous wave equations with nonlinear white noise.In Chapter 3,I prove the existence and uniqueness related weak pullback mean random attractors for mean stochastic dynamical systems generated by the non-autonomous BBM equations with nonlinear white noise.In Chapter 4,I summarize the main contents of my dissertation and point out the problems that can be further discussed.