Existence and stability of curved fronts for spatially periodic combustion reaction-diffusion equations in $\mathbb{R}^N$

TL;DR

This paper proves the existence, uniqueness, and stability of curved, polytope-shaped fronts in spatially periodic combustion reaction-diffusion equations in multi-dimensional spaces, assuming pulsating fronts exist in all directions.

Contribution

It establishes the existence and stability of curved fronts with polytope shapes in periodic media, extending the understanding of reaction-diffusion front behavior in higher dimensions.

Findings

Existence of curved fronts with polytope-like shape in $\\mathbb{R}^N$

Uniqueness of the curved front solution

Asymptotic stability of the curved fronts

Abstract

This paper is concerned with curved fronts of combustion reaction-diffusion equations in spatially periodic media in $\mathbb{R}^N$ $(N\geq2)$. Under the assumption that there are moving pulsating fronts for any given propagation direction $e \in \mathbb{S}^{N-1}$, and by constructing suitable super- and sub-solutions, we prove the existence of a curved front with polytope-like shape in $\mathbb{R}^N$. Then we show that the curved front is unique and asymptotically stable.