Stability of nonmonotone critical traveling waves for spatially discrete reaction-diffusion equations with time delay

This paper is concerned with the existence and stability of critical traveling waves (waves with minimal speed c = c(*)) for a nonmonotone spatially discrete reaction-diffusion equation with time delay. We first show the existence of critical traveling waves by a limiting argument. Then, using the technical weighted energy method with some new variations, we prove that the critical traveling waves phi(x + c(*)t) (monotone or nonmonotone) are time-asymptotically stable when the initial perturbations are small in a certain weighted Sobolev norm.