The Cauchy Problem In The 3 Thermoelasticity

In this thesis, we study the Cauchy problem in the 3D thermoelasticity with second sound,namely, the global existence and the optimal decay rates of the solution to the 3D thermoelasticity with second sound. It describes the communication process of elastic wave in nonlinear thermoelastic effect. It has important physical applications in the fields of aerodynamics,thermodynamics and the material science. The thesis is divided into the following five chapters roughly.Firstly we give a brief introduction to the physical background in the first chapter, then it leads to the 3D system by one-dimensional thermal elastic system. We also give the physical background and recent development for this research problem, as well as some of the difficulties encountered in the process of doing this article. To justify the the Cauchy problem in the 3D thermoelasticity, we analyze the main difficulties that is encountered and give the corresponding methods, namely, pure energy method with interpolation. Finally, the main parts of the article are given.In order to make full preparation for the proof of the theorem, we review the definition of Sobolev spaces and Besov spaces as well as some useful properties corresponding to each space in the second chapter. Then we present main results of the thesis.In the third chapter, we consider with the 3D thermal elastic model, we use the pure energy method with interpolation to get the nonlinear energy estimates. we make energy estimation on the dissipative variable firstly, then do gradient estimation on the degradation variable and do the Besov estimation finally. From the estimations above, we obtain the differential inequalities, and complete the proof of the main results.We give some analysis tools used in the theorem proving process in the fourth chapter.In the fifth chapter, we present the further prospect on the problem.