Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay

This paper is concerned with traveling waves for temporally delayed, spatially discrete reaction-diffusion equations without quasi-monotonicity. We first establish the existence of non-critical traveling waves (waves with speeds c > c * , where c * is minimal speed). Then by using the weighted energy method with a suitably selected weight function, we prove that all noncritical traveling waves Φ ( x + ct ) (monotone or nonmonotone) are time-asymptotically stable, when the initial perturbations around the wavefronts in a certain weighted Sobolev space are small.