Kinetic Analysis Of Two Types Of Reaction-diffusion Systems

In modern science,the reaction-diffusion equation has been widely used to describe various phenomena in physics,chemistry and biology.For example:the motion law of fluid flow in porous media,Belousov-Zhabotinski reaction,interactions and growth rules among biology groups,etc.In this paper,by using the nonlinear analysis theory,theories and methods in partial differential equation,we discuss some corresponding properties of solutions of two reaction diffusion equations.The main contents are as follows:In Chapter 1,we introduce the related backgrounds and research results of cross-diffusion model with Gray-Scott model and then give a brief introduction of this paper.In Chapter 2,a predator-prey model with cross-diffusion and homogeneous Neu-mann boundary condition is researched by using the spectrum analysis method and the degree theory.Firstly,the stability of positive constant solution is established by means of spectrum analysis method;Then the a priori estimate for the positive solutions of the steady-state problem is given by using the Maximum Principle and the Harnack Inequality;Finally,the non-existence and existence of non-constant positive solutions are analyzed by employing the energy integral method and the degree theory.Different from the pre-existing models with cross-diffusion,the mod-el in this paper is given by adding the cross-diffusion to the equation of the prey,which means that the prey runs away from the predator.The result shows that for fixed cross-diffusion coefficients,the predator and prey can coexist when the growth rates of them satisfy some conditions.In Chapter 3,we study a generalized Gray-Scott model under homogeneous Neumann boundary condition by use of bifurcation theory and the calculation of the fixed point index.Firstly,we give a priori estimate of positive solutions;Secondly,we construct a local bifurcation branch from the positive constant solution by using the local bifurcation theory;Moreover,making use of global bifurcation theory and degree theory,it is proved that the local bifurcation branch can be extended to global bifurcation branch;Finally,the stability of bifurcation solution is analyzed by the stability theory.Different from many other Gray-Scott models,the model shown in this paper is the generalized form of the Gray-Scott model.The result shows that when the diffusion rate satisfies some conditions,the concentration of reactants no longer changes and the chemical reaction reaches balance in the end.