Weak Pullback Attractors Of F-N Lattice System And Boussinesq Lattice System With Nonlinear Noise

At present,many famous scholars at home and abroad have done a lot of research work on stochastic dynamical system,but in these studies,the white noise of stochastic lattice system is usually linear additive or linear multiplicative.Because the stochastic equations with additive or multiplicative white noise can be transformed into deterministic equations for research.However,there is no method to transform stochastic equation with nonlinear noise into deterministic equation at present.Inspired by the thoughts of Wang Bixiang's literature published in JMAA in 2019,this thesis studies the asymptotic behavior of solutions of nonautonomous Fitzhugh-Nagumo(F-N)lattice system with nonlinear noise and nonautonomous Boussinesq lattice system with nonlinear noise by using the concept of weakly pullback mean random attractor.When the nonlinear noise satisfies the locally Lipschitz,we proved the existence and uniqueness of the weak pullback attractor for the mean dynamic system generated by solution operators,and finally the property of the invariant measure of the systems is proved.Weak pullback mean random attractor is a family of weakly compact sets that are attractive to arbitrary bounded sets,is an effective method to study the long-term behavior of the solution of the dynamical system.In the first chapter,we introduce the research background of stochastic dynamical systems and the main work,methods of this dissertation,then the main definitions and common inequalities involved in this dissertation are given.In the second chapter,we prove the existence of weak pullback random attractors of non-autonomous F-N lattice system with nonlinear noise and the properties of invariant measures of random attractors.In the third chapter,we prove the existence of weak pullback random attractors of non-autonomous Boussinesq lattice systems with nonlinear noise and the properties of random attractor invariance measures.In the forth chapter,we summarize the main content of this thesis,and give questions that need further consideration.