The Dynamics Of Two Classes Of Non-autonomous Diffusion Equations

In this doctoral dissertation,we consider the long-time behaviors of solutions for both non-autonomous diffusion equations with dynamical boundary conditions and non-autonomous fractional diffusion equations,respectively.We present new apriori estimates for these two classes of equations and we get a series of new and meaningful results.This thesis consists of six chapters.In Chapter 1,we firstly recall briefly the background and major results on the theory of dynamical systems.Then we introduce the background of models considered in this paper,and we also state the main contribution of this paper.In Chapter 2,we give some preliminary results that we will use throughout this paper.In Chapter 3,we study the long time behavior of solutions of a class of non-autonomous reaction diffusion equations with dynamical boundary conditions.Firstly,we establish a higher-order integrability of the difference of two solutions near the initial time.Then we prove that the?L²× L²,L²× L²?pullback ?-attractor can actually pullback attract each bounded subset in L^2+?× L^2+?-norm for all ? > 0.Moreover,we prove that the pullback ?-attractor can pullback attract each bounded subset in H¹× H^1/2-norm.In Chapter 4,we study the long time behavior of solutions of a class of nonautonomous p-Laplacian equations with dynamical boundary conditions.Firstly,we present a higher-order integrability of the difference of two solutions near the initial time.Then we prove that the?L²× L²,L²× L²?pullback ?-attractor can actually pullback attract each bounded subset in L^2+?× L^2+?-norm for any? > 0.In Chapter 5,we study the long time behavior of solutions of a class of nonautonomous fractional diffusion equations in RN.Firstly,we recall several techniques in fractional Sobolev space and then prove a preliminary lemma that we will use throughout this Chapter.Then we prove the existence of L² pullback ?_?-attractors and establish a higher-order integrability of the difference of two solutions near the initial time.In section §5.3,we prove that the L² pullback ?_?-attractor can actually pullback attract each bounded subset in L^2+?× L^2+?-norm for any ? > 0.Moreover,we prove that the pullback ?-attractor can pullback attract each bounded subset in H^s-norm.As a corollary,we establish the existence of a pullback ?-attractor in H^s.In Chapter 6,we end this thesis by listing some problems that we will consider in the future.