Stability Of Traveling Waves For Reaction-Diffusion Equations With Time Delay

In recent years,many researchers have paid more and more attention to the study of reaction-diffusion equation,and the traveling wave solution of reaction-diffusion equation is an important branch.Traveling wave solutions can not only reveal many important properties of the equation itself,but also can be a good description of the population invasion,disease transmission,combustion and other phenomena.In this dissertation,we first study the existence,uniqueness and stability of non-critical traveling waves for a class of spatially discrete reaction-diffusion equations.By constructing monotonically auxiliary equation and applying Schauder's fixed point theorem,the existence of non-critical traveling waves is obtained.Then,by choosing the appropriate weighted function,we use the technical weighted energy method to prove the asymptotic stability of all non-critical traveling waves(waves with speeds c>c*,c*is minimal speed).Secondly,we study the stability of non-critical traveling waves for a class of non-local spatially discrete reaction-diffusion equations.By taking the energy conversion method and using the nonlinear Halanay's inequality,we can prove the exponen-tial stability of all non-critical traveling wavefronts,when the initial perturbations around the wavefront is sufficiently small.Finally,we investigate the global stability of non-critical traveling waves for a class of reaction-diffusion equations with nonlocal time delay.Combining the weight-ed energy method and the comparison principle,the global exponential stability of all non-critical traveling wavefronts(includes those slow waves)is established,when the initial perturbations around the wavefront exponentially decays to zero in space as x ?-?,but it can be allowed arbitrary large in other locations.