Qualitative Study Of Solutions For A Cooperation Reaction Diffusion Model

In this paper, we mainly introduce a kind of cooperating reaction-diffusion model with p-Laplace. Since the reaction-diffusion model was set up by Shigesada, et al, the corresponding theory has been developed rapidly. We here mainly consider the case of p ?2, which diffusion terms are nonlinear and degenerated. By way of studying the analysis and understanding in previous researches, we first establish the equation by regularization, then get the properties of regularized systems, finally we give the existence of solutions and the asymptotic behavior of the original equation.In chapter 1, we introduce the background of topic, research significance and research status.In chapter 2, we lead in some basic definitions and theorems, such as gradient, divergence, some space definition, the comparison principle in classical solution, and so on. These theories provide a solid theoretical basis.In chapter 3, we chiefly introduce the regularized systems. At first, we give the definitions of the upper and lower solutions. Next, we give some appropriate assumptions, in order that we build comparison principle and iterative equation respectively. Via the upper and lower solution, the initial iteration, were built up, we could get two iterative sequences, and could prove that the two sequences are monotone bounded. According to the existence of the limits in these two sequences, we get the existence of the solutions by regularization in parabolic problems. At the same time, we could also obtain the maximum and minimal solutions of elliptic problems, and obtain asymptotic behavior of the parabolic problems.In chapter 4, due to the original problems do not have the classical solution, therefore, we discuss the existence of weak solutions. Based on limit order preserved, we put the regularization of the original problem based on that of regularized systems.