On The Study Of Solutions For Nonlinear Reaction-diffusion Equation And Its Stationary Problem

In this thesis, we study the properties of solution for nonlinear reaction-diffusion equation and its stationary problem. Nonlinear reaction-diffusion equa-tion involved in a number of issues from the physics, chemistry and mathematical model of the biological field, with a strong practical background. Thus, it has scientific significance and potential applications of putting forward all kinds of problem for nonlinear reaction-diffusion equation in mathematics and other natu ral sciences. We will study on the existence of global solution, la rge time behav-ior for several classes of nonlinear reaction-diffusion equation and the stability of solutions for stationary problem of those nonlinear reaction-diffusion equation in this article. The main contents are organized as follows:In chapter1, we state the background and development of nonlinear reaction-diffusion equation and its stationary problem, and the main work of this article.In chapter2, we consider the following Robin problem: Using upper and lower solution method, variation method, we prove the multiplic-ity of solution for problem (1). Specifically, suppose f(x) satisfies some condition, there exist a positive number βf*such that problem (1) has no positive solution when βε (0,βf*), and has at least two positive solutions when β≥βf*, of which there is a unique minimal solution.Chapter3deal with the existence of global solutions and large time asymptotic behavior of a class of nonlinear reaction-diffusion equation with Robin boundary condition. We concern for any given initial value, whether global solutions tends to an equilibrium state or not when the time tends to infinity. By the a priori estimate of global solution and the intersection property for solutions of stationary problem, we obtain that the minimal stationary solution is stable, whereas, any other stationary solution is an initial datum threshold for the existence and non-existence of its global solutions.In chapter4. we study the behavior of solutions for Dirichlet problem of non-linear homogeneous and non-homogeneous reaction-diffusion equations, the nonlin-ear reaction-diffusion equations with Robin boundary condition in bounded closed region of R" respectively. We generalize threshold results of a single equation in chapter3to the threshold of equations and obtain a threshold result when the nonlinear index meet certain conditions.Chapter5concerns a class of uneven spatial distribution of the predator-prey model, we prove the accurate a priori estimate of the positive equilibrium solution of the upper and lower bounds. Thus, the existence and nonexistence of noncon-stant positive equilibrium solution by topological degree theory and comparison principle of elliptic equations.