Researches On The Global Well-posedness Of The One-dimensional Thermodiffusion System And Related Models

Thermodiffusion system describes the process of thermodiffusion in a solid body, such a system is established based on the fundamental theories of thermodiffusion. This dissertation is concerned with the initial-boundary value problems for the global well-posedness of the one-dimensional thermodiffusion system and related models. We obtain the existence of global solutions, asymptotic stability and the global(uniform) attractors for the thermodiffusion system of type I, the thermodiffusion system with second sound, and the thermodiffusion system of type â…¢.This dissertation is divided into five chapters.Chapter1is preface, in which we mainly introduce the background and the pre-vious results, our main work, the fundamental theories for semigroup approaches, the fundamental concepts and existence theorems for global(uniform) attractors, and some classic inequalities.In Chapter2, we consider the initial-boundary value problems for the one-dimensional thermodiffusion system in a bounded region. Using the semigroup approaches and the multiplier methods, we obtain the global existence and asymptotic behavior of so-lutions for homogeneous, nonhomogeneous and semilinear thermodiffusion system subject to various boundary conditions, respectively. At the end, we get the existence of uniform attractors for the one-dimensional non-autonomous thermodiffusion sys-tem in a bounded region by establishing the uniformly asymptotic compactness of the family of processes generated by the global solutions. The main advantage of this method is that we only need to verify compactness condition with the type of energy estimates same as those for establishing the absorbing set.In Chapter3, we consider the initial-boundary value problem for a one-dimensional linear model of thermodiffusion with second sound in a bounded region. Using the semigroup approached and the energy methods we obtain the global existence and exponential stability, subject to two boundary conditions (i.e., the rigidly clamped medium with zero heat and diffusion flux; the rigidly clamped medium with tem-perature and chemical potential held on the boundary), respectively. Moreover, the existence of global attractors is established by using the method of contractive func-tions.In Chapter4, we first establish the global existence and exponential stability for the solutions to the initial-boundary value problem for the one-dimensional model of thermodiffusion of type III by the semigroup approach and the energy method. Fur-thermore, we prove the existence of global attractors by the method of contractive functions.In Chapter5, we summarize of the results of the dissertation, and predict the work in the future.