The Dynamics Of Traveling Wave Solutions For Two Types Of Reaction Diffusion Systems With Nonlocal Delay

In the past ten years,traveling wave solutions of reaction-diffusion systems with nonlocal delays have attracted extensive attention of scientists.In mathematical theory,traveling wave solution can be used as the steady state solution of a reaction diffusion equation,which plays an important role in studying the asymptotic behavior of solutions to initial value problems of partial differential equations.In real life,traveling wave solutions can explain the finite velocity propagation and oscillation phenomena in nature.In the study of traveling wave solutions,stability is one of the hot topics.In this paper,we will focus on the stability of traveling wave solutions for two kinds of reaction diffusion systems with nonlocal delays.In chapter 2,we study the existence and global exponential stability of traveling wavefronts for a class of Belousov-Zhabotinskiil reaction-diffusion systems with nonlocal delays.First,by constructing a pair of appropriate upper and lower solutions and then using the Schauder fixed point theorem,the existence of a monostable traveling wavefront is proved.Then,applying the weighted energy method and the comparison principle,it is proved that when the initial perturbation around the wavefront solution is in a weighted Sobolev space,the solution of the corresponding Cauchy problem converges exponentially to the monostable traveling wavefront in time.In chapter 3,the stability of the traveling wavefronts of a food finite population model with nonlocal delay is studied.First,using the standard asymptotic theory,the asymptotic behavior of the traveling wavefronts at both ends of infinity is obtained.Then,through the spectral analysis of the linearization operator,it is proved that the traveling wavefront is unstable when the initial perturbation of the traveling wavefront is in the bounded continuous function space.When the initial perturbation of the traveling wavefront belongs to some exponential weighted space,the traveling wavefront is asymptotically stable.