Qualitative Studies Of Some Parabolic Equations

Studying nonlinear phenomena appearing in various fields such as physics, chemistry, biology and economics, etc, has become an important aspect in the field of non-linear partial differential equation. Three kind problems are discussed in the present paper: one is regularities of weak solutions to a fast diffusion equation, mainly concerning universal Holder continuity and uniform Holder estimates; one is the global existence of weak solutions to a prey-predator model with strong cross-diffusion; the other one is blow-up properties of solutions to several non-linear parabolic equations, which include conditions of global existence (namely, critical blow-up exponent), blow-up rate estimates and blow-up set, etc. The outline of this paper is as follows.Charpter 1 is the preface, recalling the background of the related work, and summarizing the main work of the present paper.Charpter 2 deals with regularities of weak solutions to a fast diffusion equation. The main feature of the equation is that its parameters are only assumed to be measurable. By introducing a generalized De Giorgi class and redefining weak solutions (weak super- and sub-solutions), the relationship between them is established. Then by applying measure arguments, we demonstrate that all weak solutions are universal Holder continuous, and uniform Holder estimates are also presented.The subject of Charpter 3 is a parabolic prey-predator model with strong cross-diffusion. By using of finite differences and an entropy inequality, global existence of a weak solution is assumed in multidimensional space. Furthermore, we illustrate that this solution is non-negative.Charpter 4 concerns a parabolic system defined in half-space and coupled with two nonlinear reaction terms and two nonlinear boundary conditions. By introducing a system of suitable elementary inequalities, and by virtue of scaling arguments and iteration methods, blow-up rate estimates and blow-up set of solutions are obtained. In particular, for the case of one dimensional space, a complete conclusion about blow-up rate estimates and blow-up set is established.Charpter 5 deals with blow-up propertities of positive solutions to a quasilinear parabolic equation. We firstly obtain critical blow-up exponent by constructing super-solutions and sub-solutions, and then describles blow-up set by the comparison principle. Finally, application of scaling analysis gives blow-up rate estimates.