Pulsating Fronts of Spatially Periodic Bistable Reaction–Diffusion Equations Around an Obstacle

In this paper, we study a spatially periodic bistable-type reaction–diffusion equation in so-called exterior domains Ω = R^N\ K , where K⊂ R^N is a compact set and denotes an obstacle. For any direction e∈ S^N-1 , if the spatially periodic bistable reaction–diffusion equation in R^N admits a moving pulsating front (i.e., the wave speed is nonzero), we first prove the existence and uniqueness of entire solution in the exterior domain Ω , which is emanated from the moving pulsating front. Assuming further that the propagation of the entire solution is complete (i.e., convergence to 1), we prove that the entire solution is a transition front connecting 0 and 1 and is trapped between two translates of the moving pulsating front as time goes to + ∞ . In particular, applying a Liouville-type result, we prove that the entire solution can eventually recover to the same moving pulsating front after crossing the obstacle K by providing some appropriate hypotheses.