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

TL;DR

The paper addresses existence, uniqueness, and stability of curved fronts for spatially periodic combustion reaction-diffusion equations in $\mathbb{R}^N$ by assuming pulsating fronts exist in every direction. It develops a sub- and supersolution framework augmented by constructing a polytope-like hypersurface $\Sigma$ that prescribes the front's asymptotic geometry, and proves the curved front $V(t,z)$ is unique and asymptotically stable with a polyhedral boundary $\Gamma_t=\partial\mathcal{Q}+\hat{c}te_0$. The results extend the understanding of non-planar traveling fronts in periodic media, showing curved fronts can be rigorously constructed and are robust to perturbations, with convergence to pulsating fronts along each face. The work leverages pulsating-front theory, a hypersurface construction, and a detailed comparison/monotone limit strategy to establish a globally stable curved front that acts as a transition front between the equilibria $0$ and $1$.

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.

Theorems & Definitions (19)

• Definition 1.1
• Definition 1.2
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Definition 2.6
• Lemma 2.7
• Lemma 2.8