Well-posedness Of Two Kinds Of Parabolic Hyperbolic Equations

The present PH. D. dissertation is mainly devoted to study two kinds ofparabolic and/or hyperbolic equations. One model is degenerate parabolic-hyperbolicequations. The other model is the equations of thermoelasticity.Degenerate parabolic-hyperbolic equations arise in lots of physical models,such as porous medium flow, pollutant transportation phenomenon, sedimentation-consolidation process, financial decision etc. The main feature of this type ofequations lies in that whether the equation is parabolic or hyperbolic depends onthe solutions itself. So, it is very difcult to study the qualitative behavior of thesolutions.Thermoelasticity is the subject which study the behavior of strain and stressunder the influence of temperature for elastic, heat conductive media. The equa-tions of thermoelasticity is established by using heat conducting process and thefirst and the second law of thermodynamics under the framework of elasticity.When the heat conducting process is described by Fourier’s Law, the correspondingequations are called classical thermoelasticity. It is a parabolic hyperbolic coupledsystem. When the heat conducting process is described by Cattaneo’s Law, thecorresponding equations are called thermoelasticity with second sound. It is a fullyhyperbolic system.The basic knowledge for both parabolic and hyperbolic equations are wellneeded in studying these two types of equations. Our main results are describedas follows:1. We study the Neumann boundary value problem for general isotropic de-generate parabolic-hyperbolic equations and give its well-posedness. We give thedefinition of entropy solutions for Neumann type problem. The main feature inthis definition is that we present a new boundary condition. This new boundary condition is based on the strong trace result which we give for general isotropic degenerate parabolic-hyperbolic equations. Once a proper boundary condition is given, the methods of Kruzkov’s doubling of variables as well as boundary layer se-quences can be used to obtain the uniqueness result. But there are two hypothesis for the proof of uniqueness. One is the restrictions on the flux and diffusive func-tions (the nonlinearity-diffusivity condition), the other is that the domain must be a bounded rectangle. Even though, our theorem still first give a uniqueness results for multidimensional Neumann type problem for isotropic degenerate parabolic-hyperbolic equations. The existence of entropy solutions is also discussed.2. We investigate the formation of singularities in thermoelasticity with sec-ond sound. Transforming into Euler coordinates and combining ideas from T. C. Sideris [72], used for compressible fluids, and M.A. Tarabek [77], used for small data large time existence in second sound models, we are able to show that there are in general no global smooth solutions for large initial data. In contrast to the situation for classical thermoelasticity, we require largeness of the data itself, not of its derivatives.