Stability of pulsating fronts for bistable reaction-diffusion equations in spatially periodic media

Abstract

The first part of the paper is concerned with the asymptotic stability of pulsating fronts in RN for spatially periodic bistable reaction-diffusion equations with respect to decaying perturbations. Precisely, we show that the solution u(t,x) of the initial value problem converges to the pulsating front as t→+∞ uniformly in RN. In the second part, we investigate the existence and asymptotic behaviour of the entire solution u(t,x) emanating from a pulsating front for the equation in exterior domains. The proof of the asymptotic behaviour is relying on the application of the proof for the stability of the pulsating front.