Analysis Of The Solutions Of Two Classes Of Reaction Diffusion Equation

In recent years, model has played a vital role in physics, biochemistry, medicine and some newly developing natural subjects whose practical problems can be re-solved. It helps scientists to simulate certain experiments and describe some natural phenomena. After a large number of models have been studied, it is found that a majority of them are derived from reaction-diffusion equations. With these equations fully investigated, some natural and biological phenomena can be more scientifical-ly illustrated, predicated and prevented, on the one hand; on the other, equations theories have been improving with more intensive researches done.The thesis focuses on two types of reaction-diffusion models:one is a Gause-type predator-prey model with cross-diffusion the other is a viral dynamics model with diffusion and B-D functional response The theories adopted are nonlinear analysis and nonlinear partial differential equa-tions, especially those of parabolic equations and corresponding elliptic equations.The thesis consists of two chapters.In chapter1, a Gause-type predator-prey model with cross-diffusion under ho-mogeneous Neumann boundary condition is investigated. It contains three parts:in part one, a priori estimate for upper and lower bounds of postive solution is dis-cussed by means of the maximum principle and Harnack inequality; in part two, the non-existence of the non-constant postive solution is proved with the help of the intergral property and two important inequalities (e-Yong inequality and Poincare inequality); and in part three, the existence of the non-constant postive solution is proved through the priori estimate and Leray-Schauder degree theory.In chapter2, coexistence states of a viral dynamics model with diffusion and Beddington-DeAngelis functional response under homogeneous Neumann boundary condition is studied. It is also made up of three parts:first, a priori estimate is ob-tained with the help of linearization theory; second, the local asymptotical stability is discussed through Hurwitz theorem; and third, the global asymptotical stability of disease-free equilibrium solution is proved by constructing upper and lower solutions and its associated monotone iterations.