Stability Of Traveling Waves For Spatially Discrete Reaction-Diffusion Systems

In recent years,the spatially discrete reaction-diffusion system has attracted the attention of many scholars,since the spread of infectious diseases,the ecology evolution of interspecific relationship and the atmospheric diffusion are all the spa?tially discrete diffusion.Traveling wave solution is the most important topic among the researches of this system.We will further talk about the stability of traveling waves for spatially discrete reaction-diffusion systems.The main result is divided into three sections.In section 2,for a spatially discrete reaction-diffusion equation with time delay,we firstly show the existence of critical traveling waves by a limiting argument.Then using the technical weighted energy method with Young-inequality,we prove that the critical traveling waves ?(x + c*t)(monotone or non-monotone)are time-asymptotically stable,when the initial perturbations are small in a certain weighted Sobolev norm.In section 3,for a discrete diffusive Lotka-Volterra competition system,using the comparison principle and the weighted energy method,we prove that the traveling wavefronts under the large wave speed are exponentially stable,when the initial perturbation around the traveling wavefronts decays exponentially as x ?-?,but can be arbitrary large in other locations.In section 4,for a lattice differential Lotka-Volterra competition system,by the same methods as those of in section 3,we obtain the exponentially stability of the traveling wavefronts under the large wave speed.