Propagation phenomena of spatially periodic combustion reaction-diffusion equations around an obstacle

Abstract

This paper is concerned with propagation phenomena of spatially periodic combustion reaction-diffusion equations in exterior domains. It is known that there is a pulsating front connecting 0 and 1 with positive speed in ${\mathbb{R}}^N$ for any direction $e\in {\mathbb{S}}^{N-1}$. We first prove that there exists an entire solution originating from a pulsating front in the exterior domain. Then, we prove that the entire solution propagates completely. Additionally, by constructing appropriate super- and sub-solutions, we establish that the entire solution is a transition front connecting 0 and 1, and that it is trapped between two translates of the pulsating front as $t\to +\infty$. Finally, under a suitable assumption, we show that the entire solution converges to the same pulsating front as $t\to +\infty$, as well as the uniqueness of such entire solutions.