Asymptotic Dynamics Of Nonlinear Evolution Equations Under Random Disturbance

Random attractor is a very useful mathematical tool for studying the long-term asymptotic behavior of random dynamical systems.In fact,the essence of a random attractor is a compact invariant set,and the invariant set varies with time,which is of great significance for understanding the local and global dynamical characteristics of the system.Therefore,the study of random attractor theory for random dynamical systems is not only a good extension of the classical theory of global attractor for deterministic dynamical systems,but also has been successfully applied to many important stochastic partial differential equations.In this thesis,the asymptotic behavior of the non-autonomous Ginzburg-Landau equation with random perturbation is studied from three aspects:colored noise and white noise,thin domain and fixed domain,and random initial data by using functional analysis and infinite dimensional dynamical system theory.In addition,the dynamical behavior of stochastic fractional nonlocal reaction-diffusion equations driven by additive white noise is studied.The details are as follows:Firstly,we consider the asymptotic behavior of the non-autonomous random Ginzburg-Landau equation driven by nonlinear colored noise on unbounded domains.Since the noise is nonlinear and the Wiener process W is almost everywhere non-differentiable,we consider its approximation equation.By using the classical Galerkin approximation method and the convergence estimates of the solutions,the well-posedness of the weak solutions of the nonautonomous random Ginzburg-Landau equation driven by nonlinear colored noise is proved,and then a continuous non-autonomous random dynamical system is generated.It is worth noting that the Sobolev embedding on unbounded domains is non-compact,we use the uniform tail estimates method combined with the property of colored noise to overcome this difficulty,thus prove the existence and uniqueness of pullback random attractors.In particular,for the case of linear multiplicative colored noise,we show that its pullback random attractor converges to the pullback random attractor of the non-autonomous random Ginzburg-Landau equation driven by linear multiplicative white noise.Secondly,we investigate the asymptotic behavior of the non-autonomous random Ginzburg-Landau equation with nonlinear colored noise on unbounded thin domains.Since the thin domain is variable,we first transform the equation from an unbounded thin domain to a fixed unbounded domain,and then prove that there exists a unique weak solution to the equation,and the continuous random dynamical system generated by the weak solution has a unique pullback random attractor.Since the thin domain is dependent on the thickness parameter,we prove the upper semi-continuity of the pullback random attractor with respect to the parameter by using the property of the average function.Then,we study the weak mean dynamics of complex valued GinzburgLandau equations with random initial values.By using the weak pullback mean random attractor theory,we prove the well-posedness of the solution of the Ginzburg-Landau equation with random initial values on unbounded domains,so as to establish the mean random dynamical system.Then,the existence,uniqueness and periodicity of the weak pullback mean random attractor are proved.In addition,we extend the existence,uniqueness and periodicity of the weak pullback mean random attractor to the weighted space L2(Ω,Lσ2(R))with the hypothesis of the nonlinear drift term weakened.Finally,we are concerned with the dynamics of stochastic fractional nonlocal reaction-diffusion equations with additive white noise.For the stochastic equation disturbed by linear white noise,we use the usual transformation method to transform the equation into a pathwise random equation.However,new non-local terms are appeared in the transformation process,we construct Caratheodory function,and prove the existence of local solutions by using the extended Peano’s theorem.By deriving the uniform estimates,the existence and uniqueness of the weak solution are obtained.The existence and uniqueness of pullback random attractors are obtained by proving the existence of pullback absorbing set and the pullback asymptotic compactness of random dynamical systems.In addition,for the case of additive colored noise,we prove the existence and uniqueness of the weak solution and the pullback random attractor by a similar method,and further prove the upper semi-continuity of the pullback random attractor as δ approaches zero.