Bistable Pulsating Fronts for Reaction-Diffusion Equations in a Periodic Habitat

This paper is concerned with the existence and qualitative properties of pulsating fronts for spatially periodic reaction-diffusion equations with bistable nonlinearities. We focus especially on the influence of the spatial period and, under various assumptions on the reaction terms and by using different types of arguments, we show several existence results when the spatial period is small or large. We also establish some properties of the set of periods for which there exist nonstationary fronts. Furthermore, we prove the existence of stationary fronts or non-stationary partial fronts at any period that is on the boundary of this set. Lastly, we characterize the sign of the front speeds and show the global exponential stability of the non-stationary fronts for various classes of initial conditions. Our arguments are based on the maximum principle, spectral analysis, and dynamical systems approach.