Qualitative Study On The Solutions Of Two Nonlinear Reaction-diffusion Equations

Reaction-diffusion phenomenon can be widely found in natural world and reaction-diffusion equations have widespread applications in modern science and technology.The research of reaction-diffusion equations mainly focuses on the spatial distribution and diffusion law as well as the effect of time and space on the diffusion for a natural system,which guarantees a better understanding upon the influence of the diffusion rate on surrounding environment.The most typical reaction-diffusion models are the predator-prey model in biology and the Sel'kov model in chemistry.There are several ways to discuss reaction-diffusion equations,such as the method of upper and lower solution,topological degree theory and the energy estimates in partial differential theory,the applications of which give a better understanding on the properties and behaviors of solutions to nonlinear reaction-diffusion systems.In this thesis,we are devoted to conducting the qualitative study on positive solu-tions of two reaction-diffusion equations by analyzing the existence,nonexistence and asymptotic stability of non-constant positive solutions or traveling wave solutions.The full text includes the following three chapters:Chapter 1 introduces the background of reaction-diffusion equations,and a brief history of traveling wave solutions in predator-prey models,asymptotic behavior,Sel'kov model,saturation law and the main work of this work.Chapter 2 considers an n-dimensional diffusive predator-prey systems.By us-ing upper and lower solution method and Schauder's fixed point theorem,we prove the existence and find the minimal wave speed of traveling wave solutions.Through asymp-totic analysis technique,we derive the asymptotic behaviors of traveling wave solutions,which solve the open problem mentioned in 2014.Meanwhile,we extend the original three dimensional systems to arbitrary finite dimensional(n-dimensional)systems,and complement the results in references.Chapter 3 discusses the Sel'kov model with saturation law.By applying the im-plicit function theorem and Leray-Schauder topological degree,we prove the stability,existence and nonexistence of non-constant steady states.Moreover,there is a new phe-nomenon that it is the saturation law to determine the formation of spatial patterns:Tur-ing pattern may occur when the saturation coefficient is small but will not occur when the coefficient becomes large.