A Study On Steady States Of Two Classical Reaction-diffusion Systems

Abstract: This dissertation is a study on steady state solutions of two classical reaction-diffusion systems. The main contents are as follows:In the first chapter, we briefly introduce the background, research sig-nificace of this thesis and the mathematical notations used throughout this paper.In the second chapter, we prove the non-existence of non-constant steady state of the Chemotaxis model by applying the Maximun Principle, Implicit Function Theorem and Finite Covering Theorem. We establish the critical value of the chemotactic coefficient between the existence and the non-existence of non-constant steady states. This work is a supplement and improvement over known results.In the third chapter, we consider the non-constant steady state solutions of a Brusselator model. We freak for the first time the concentration of one intermediary reactant as a bifurcation parameter to study the local and global structure of the non-constant steady state of the Brusselator model by applying local and global bifurcation theory. Through asymptotic analysis, we derive an explicit formula for the non-constant steady states.In the fourth chapter, based on the results obtain in the third chapter, we establish the stability criteria and find a selection mechanism of the wave modes for the non-constant steady states by computing the leading term of the principal eigenvalue. Our results demonstrate that all of the pick bifurcations other than the one at the first bifurcation location are unstable. Simulations for specific model parameters are presented to illustrate our analytical results.