Stability Of Traveling Wave Solutions For Two Types Of Reaction-Diffusion Systems

In recent decades,the traveling wave solutions of reaction-diffusion systems have attracted more attention from researchers.The traveling wave solutions are special translation invariant solutions.In the study of mathematical theory,the traveling wave solutions can reveal many important properties of equations.In practical applications,the traveling wave solutions can explain the propagations with finite speed and oscillatory phenomena in nature.In the study of traveling wave solutions,the stability is an important topicFirstly,we study a delayed reaction-diffusion system without quasi-monotonicity.By constructing auxiliary equations and using Schauder's fixed point theorem,we show the existence of traveling wave solutions.Then taking the Fourier transfor-m and the weighted energy method,we prove that the traveling wave solutions(monotone or non-monotone)are globally exponentially stable in L?(R)with the exponential convergence rate t-1/2e-?t‘for some constant ?>0Secondly,we investigate a nonlocal dispersal Lotka-Volterra weak competitive system.This system admits a traveling wavefront connecting a boundary equilib-rium and a co-existence equilibrium.By using the comparison principle and the weighted energy method,we prove that the traveling wavefronts with large speed are exponentially stable in L?(R)with the exponential convergence rate e-?t for some constant ?>0.