Long-time Dynamics For Thermoelastic Bresse System Of Type Ⅲ With Infinite Memory

Recently,theories of Bresse system play an increasing role in many fields,such as engineering,physics and material science,due to its wide applications.In this work,we investigate the long-time dynamics for thermoelastic Bresse system of type Ⅲ.By using the methods developed by Chueshov and Lasiecka,we get the quasi-stability property of the system and obtain the existence of a global attractor which has finite fractal dimension.We obtain the regularity of attractors with respect to time.Result on exponential attractors of the system is also proved.This paper is organized as follows:In Chapter 1,we introduce the research background of Bresse system problem and give three types of thermoelastic equations.We summarize the research results of Bresse system at home and abroad,and expond the main research work and research results of this paper.In Chapter 2,we give some necessary basic concepts,lemmas and inequalities in common use.In Chapter 3,we study the well-posedness of Bresse system.Since the existence of the memory term in the thermal effect equation of the Bresse system,the system is a nonautonomous system.To solve this problem,we introduce a new variable to transform the system into an autonomous system.Then,we prove the well-posedness of Bresse system by using semigroup method,and determine that the solution map forms a dynamic system.In Chapter 4,we study the long-time dynamic behavior of Bresse system.If we prove the existence of absorbing set by traditional methods and verify the compactness of semigroups by the decomposition technique,it will be difficult.In order to overcome this difficulty,we use a Lyapunov functional to prove that the dynamical system is a gradient system.Then,we use a series of linear combinations of energy estimates to obtain the quasi-stability dynamical system.We obtain the existence of global attractors,the regularity of attractors with respect to time and the existence of generalized exponential attractors.