Asymptotic stability of traveling waves for delayed reaction-diffusion equations with crossing-monostability

This paper is concerned with the traveling waves for a class of delayed reaction-diffusion equations with crossing-monostability. In the previous papers, we established the existence and uniqueness of traveling waves which may not be monotone. However, the stability of such traveling waves remains open. In this paper, by means of the (technical) weighted energy method, we prove that the traveling wave is exponentially stable, when the initial perturbation around the wave is relatively small in a weighted norm. As applications, we consider the delayed diffusive Nicholson??s blowflies equation in population dynamics and Mackey?CGlass model in physiology.