Dynamical Approach To Steady-state Equations And Random Attractors For Stochastic Dynamical Systems

Dynamical systems describe the behavior of the system evolving with time in the phase space?state space?,such as the pendulum movement,the motion of galaxies and the fluid motion,etc.When the dynamical system is trivial,that is,the system tends to be static or stationary,the solution then tends to an equilibrium as time t??.In this case the system becomes a steady-state equation.Variational method is a classical tool to study steady-state equations,however,it has certain limitations.In this thesis,we adopt a dynamical approach to investigate a nonlinear elliptic equation,which provides a new idea for the steady-state problem.On the other hand,dynamical systems can be influenced by natural or man-made random factors,such as random disturbance,random environments,random initial or boundary conditions,random input,etc.Stochastic dynamical systems are the proper mathematical models of complex systems under stochastic influence.They can describe the complex phenomena that deterministic dynamical systems cannot describe,which has important practical significance.In the following,we consider the random attractors and their dynamical behaviors of some stochastic partial differential equations.This thesis contains five chapters.In Chapter 1,we introduce evolution of infinite-dimensional dynamical systems and stochastic dynamical systems.Then we elaborate our work and arrangements in this doctoral dissertation.In Chapter 2,we present some preliminary definitions and results that will be used in this thesis.In Chapter 3,we study the existence of ground states of a nonlinear elliptic e-quation with potentials may decay to 0 at infinity.Different from the traditional variational method,we use dynamical approach to get the positive solution of the problem by the global solution of the corresponding parabolic equations.Under some additional conditions,some global solutions have?-limit sets containing positive e-quilibria.In Chapter 4,we use the method of smooth approximation to examine the random attractor for two classes of stochastic partial differential equations?SPDEs?.Roughly speaking,we perturb the SPDEs by a Wong-Zakai scheme using smooth colored noise approximation rather than the usual polygonal approximation.After establishing the existence of the random attractor of the perturbed system,we prove that when the colored noise tends to the white noise,the random attractor of the perturbed system with colored noise converges to that of the original SPDEs by invoking some continuity results on attractors in random dynamical systems.In Chapter 5,we consider the long time behavior of solutions to a stochastic reaction-diffusion equation with multiplicative noise and deterministic non-autonomous forcing.Using some new techniques,we first establish some new estimates about higher-order integrability of the difference of the solutions near the initial time,based on which,then we prove that the attraction in the usual?L²?Q?,L²?Q??D-pullback random attractor indeed can be L^2+?-norm for any??[0,?)and the H₀¹?Q?-continuity with respect to initial data as well as the existence of pullback random attractor in H₀¹?Q?.