Propagating speeds of bistable transition fronts in spatially periodic media

Abstract

This paper is concerned with the propagating speeds of transition fronts in $$\mathbb {R}^N$$ for spatially periodic bistable reaction–diffusion equations. The notion of transition fronts generalizes the standard notions of traveling fronts. Under the a priori assumption that there exist pulsating fronts for every direction e with nonzero speeds, we show some continuity and differentiability properties of the front speeds and profiles with respect to the direction e. Finally, we prove that the propagating speed of any transition front is larger than the infimum of speeds of pulsating fronts and less than the supremum of speeds of pulsating fronts.