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 Omega = R-N\K, where K subset of R-N is a compact set and denotes an obstacle. For any direction e is an element of SN-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 Omega, 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 +infinity. 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.