Well-posedness And Dynamics Of Several Kinds Of Reaction-diffusion Equations

In this paper,we mainly study the well-posedness and dynamics of several kinds of reaction-diffusion equations,including non-autonomous three-component reversible Gray-Scott system,stochastic three-component Gray-Scott system and stochastic two-compartment Gray-Scott system.This paper is divided into six chapters.In chapter 1,we introduce the physical background and research status of dynamical systems,attractors and reaction-diffusion equations,and give the innovation and overall structure of this paper.In chapter 2,we define some symbols and give some definitions,theorems,propositions and properties,including the definitions of non-autonomous random dynamical system and attractors.In chapter 3,we consider the pullback attractor of non-autonomous three-component reversible Gray-Scott system on an unbounded domain.By using the uniform bounded-ness condition instead of the compactness condition,we prove the forward convergence and backward convergence of the pullback attractor to the global attractor.In chapter 4,we study the uniform attractor of stochastic three-component Gray-Scott system with multiplicative noise.We first show the existence and uniqueness of the solution for stochastic three-component Gray-Scott system by using the standard Galerkin method.Then,by proving that the corresponding conditions are satisfied,we obtain that the stochas-tic three-component Gray-Scott system can generate a non-autonomous random dynamical system.Finally,we prove the existence of the uniform attractor and the cocycle attractor by using the uniform prior estimation for the solution of stochastic three-component Gray-Scott system.In chapter 5,we study the uniform attractor of stochastic two-compartment Gray-Scott system with multiplicative noise.Firstly,we prove the existence and uniqueness of the so-lution for stochastic two-compartment Gray-Scott system by using Galerkin method.We then prove that the stochastic two-compartment Gray-Scott system can generate a non-autonomous random dynamical system.Finally,the existence of the system's uniform at-tractor and cocycle attractor is obtained by using the uniform prior estimation.In chapter 6,we not only give the summary of this paper,but also put forward some feasible ideas.