Properties Of Solutions For Nonlinear Degenerate Reaction-Diffusion Systems

Nonlinear partial differential equation is one of the most accurate mathematic models describing nonlinear phenomenon. Moreover, it is an important bridge between the theory of mathematics and practical applications. The study of reaction-diffusion model mainly resolves two basic problems: global existence and non-existence (blow up in finite time), which includes issues on the background of physics, chemistry, ecology, biology and other subjects. The properties of solutions of reaction-diffusion systems can be used to describe, explain or forecast lots of phenomenon of nature, exerting in subjects and engineering technology.This thesis is based on compare theory and the method of supersolutions and subsolutions, mainly considered the properties of solutions to two kinds of degenerate nonlinear parabolic systems in quality respectively, which includes global existence; blow up in finite time and blow-up rate estimate.In part one, the background and development of the parabolic system in the introduction is described briefly, and some basic knowledge which will be used in this thesis without proving are reviewed.In part two, this thesis is dealt with properties of solutions to a degenerate parabolic system coupled via non-local sources, subject to homogeneous Dirichlet conditions and nonnegative initial data. In the first place, the local existence of the classical solutions is proved by the regulate methods and standard parabolic theory. Secondly, the global existence and blow-up in finite time of solutions are obtained by the method of supersolutions and subsolutions. Here supersolutions and subsolutions are constructed by many function forms, such as special function, blow-up factor and solutions of ordinary differential equations.Last, the estimates of blow-up rate for the blow-up solutions and conditions of the solutions blowing up globaly are obtained.In part three, global existence and blow-up properties of solutions for a multi-coupled degenerate parabolic system with absorption and nonlinear boundary flux are discussed. Because of the nonlinear boundary flux, the construction of super and sub-solutions are more difficult and complex. Some conditions of global existence and finite time blow up for the solutions are obtained.