Research On The Related Problems Of Nonclassical Diffusion Equations With Memory In Time-Dependent Spaces

In this paper,we take a class of nonclassical diffusion equation with fading memory in time-dependent space as the research object and study the long-time dynamical behavior of its solution.We investigate the well-posedness of the equation and the existence of attractor when the nonlinearity f satisfies different conditions.In chapter 1,we provide an explanation of the research background and signifi-cance,as well as a summary of the existing research results and the main work and conclusions of this article.In chapter 2,based on the consideration of the functionε（t）and the memory term,we construct time-dependent spaces for our research.More-over,we expound relevant Definitions and Lemmas.In chapter 3,we study the case where the nonlinearity satisfies critical exponential growth.Firstly,we use Faedo-Galerkin method to find the approximate solution and make a uniform boundedness priori estimate of it,thus we are aware that the equation has a unique strong solution.Secondly,a strongly continuous process is defined and the existence of time-dependent absorbing set is obtained.Thirdly,the decomposition technique is used to prove the asymptotic compactness of the process.Finally,we acquire a strong time-dependent global attractor A={A_t}_t∈Rin M_t¹.In chapter 4,we study the case where the non-linearity satisfies arbitrary polynomial growth.Firstly,we use Faedo-Galerkin method to find the approximate solution and make a uniform boundedness priori estimate of it,thus we are aware that the equation has a unique strong solution.Secondly,a strongly continuous process is defined and the existence of time-dependent absorbing set is obtained.Thirdly,the contractive function method is used to prove the asymp-totic compactness of the process.Finally,we acquire a strong time-dependent global attractor A={A_t}_t∈Rin M_t¹.In Chapter 5,we propose a summary and outlook for this article.