Asymptotic Behavior Of Solutions Of Diffusion Equations With Memory

In this thesis,we mainly discuss two types of developmental partial differential equations,one is the diffusion equation with memory term and the other is the wave type equation with nonclassical dissipation term and memory term.Firstly,the origin and development of infinite dimensional dynamical systems are briefly described,and the main research results of this thesis are summarized.Then,in the second chapter,we briefly introduce the basic concepts,basic theories and some common inequalities.The main research content of this thesis includes two parts,which are divided into chapter 3 and chapter 4.In chapter 3,the global weak solution of diffusion equation with memory term is studied Bi-product-spaces(CL2(RN),CLp(RN))in the time dependent on back to the existence of the attractor.Since,the compact embedding theorem is not hold in unbounded domain.In addition,due to time delay g(t,ut),we studied problems of the phase space into Banach space CX,not Banach space X,so we can not directly use traditional methods to validation pullback asymptotic compactness for the process {U(t,r)}t≥τ.In order to overcome the difficulty caused by the existence of time-dependent memory variables,we first decompose the whole space RN into two parts by using the truncation function method(see Wang[31])combined with innovative analytical techniques:A ball BR with this radius R,is large enough and its complement set is BRC.It is proved that the system energy functional is sufficiently small over BRC.Then,a new operator decomposition method is used to construct the asymptotic contraction function in the sphere BR.The pullback asymptotic compactness of the solution process family is obtained,and the existence of the pullback attractor is proved.In chapter 4,we study the well-posedness of wave-type equation with nonclassical dissipative terms and memory terms,and obtain the existence and the uniqueness of the solutions and then the continuous dependence of the initial value.The function of the non-classical dissipative term lies in that sometimes it dissipates and sometimes it gathers energy,and the energy functional is essentially different from the classical one decrement.It is very difficult to estimate the system’s dissipative property even by the classical method.We prove the uniform boundness of finite dimensional approximation sequences by using the nonclassical Galerkin method and a new analytical technique.By using the weak compactness of the phase space of the system,it is found that there are corresponding weak convergent subsequences in the finite dimensional approximation sequence,and then the wave shape equation with nonclassical dissipation term and memory term is well established.