On Traveling Fronts of Combustion Equations in Spatially Periodic Media

Abstract

This paper is concerned with traveling fronts of spatially periodic reaction–diffusion equations with combustion nonlinearity in \(\mathbb {R}^N\). It is known that for any given propagation direction \(e\in \mathbb {S}^{N-1}\), the equation admits a pulsating front connecting two equilibria 0 and 1. In this paper we firstly give exact asymptotic behaviors of the pulsating front and its derivatives at infinity, and establish uniform decay estimates of the pulsating fronts at infinity on the propagation direction \(e\in \mathbb {S}^{N-1}\). Following the uniform estimates, we then show continuous Fréchet differentiability of the pulsating fronts with respect to the propagation direction. Lastly, using the differentiability, we establish the existence, uniqueness and stability of curved fronts with V-shape in \(\mathbb {R}^2\) by constructing suitable super- and subsolutions.