Blow-up For Nonlinear Diffusion Equations (Systems)

Nonlinear diffusion equations (systems) come from many mathematical models in physics, chemistry, biology and so on, which describe the heat transfer and materials diffusion. Blow-up theory of diffusion equations is the one of the important branches of partial differential equations. Global existence and blowup solutions present the stability and nonstability respectively, in heat transfer and materials diffusion. In view of mathmatical theory and applied sciences, it is very important to investigate the blow-up properties for diffusion equations. In this thesis, we will study several diffusion equations (systems) arose in applied sciences.This paper includes four chapters.In the preface of this paper, we introduce some recent important reserch results about blow-up theory of nonlinear diffusion equations, and we state our main work in this paper.In Chapter 2, we introduce a reaction diffusion system with localized sources. Under appropriate hypotheses, we obtain that the solution either exists globally or blows up in finite time by making use of super and sub solutions techniques.In Chapter 3, we deal with a fast diffusion equation with a nonlinear boundary flux. We first get the behavior of the solution at infinity and establish the critical global existence exponent and critical Fujita exponent for the fast diffusion equation, furthermore give the blow-up set and upper bound of the blow-up rate for the nonglobal solutions.In Chapter 4, we establish the critical global existence curve and critical Fu-jita curve for a degenerate parabolic system with nonlinear boundary conditions in multi-dimension.