The Study Of Solutions Of Two Classes Of Nonlinear Parabolic Problems

This dissertation is devoted to the local existence and uniqueness of the solution for a class of degenerate quasi-linear parabolic equations, and an existence result of entropy solutions to a class of nonlinear parabolic problems.The dissertation includes five parts:In Chapter 1, the basic application background, basic knowledge, advanced studies and the main idea of this dissertation are introduced.In Chapter 2, 3, we investigate the local existence and uniqueness of the solution for a class of degenerate quasi-linear parabolic equations, and in Chapter 2, we also show that under certain conditions the solution blows up. First we need some transformation of function. We transform the degenerate equations into the non-degenerate equations;then using regularity and the Schauder fixed point theorem, we apply comparison theorem and the supersolutions to obtain the local existence and uniqueness of the solutions. Finally, by dealing with the approximation of the non-degenerate equations, we get the solution of the original equations.In Chapter 4, 5, an existence result of entropy solutions to a class of nonlinear parabolic problems is established. They have different lower order term, either in Caratheodory form, or in divergence form. By introducing truncated function, we apply the integral norm and compact theorem to obtain entropy solutions. We apply Sobolev embedding inequalities, Holder inequalities, Young inequalities and some normal inequalities in this chapter.