Curved fronts of bistable reaction-diffusion equations in spatially periodic media: $N\ge 2$

TL;DR

This paper investigates the existence, uniqueness, and stability of curved transition fronts in bistable reaction-diffusion equations within spatially periodic media for dimensions two and higher, assuming the existence of pulsating fronts.

Contribution

It establishes the existence of polytope-like curved fronts with stable properties, under certain assumptions, extending the understanding of front solutions in higher-dimensional periodic media.

Findings

Existence of polytope-like curved fronts with specific zones

Reversal of zones under different conditions

Curved fronts are unique and asymptotically stable

Abstract

This paper is concerned with curved fronts of bistable reaction-diffusion equations in spatially periodic media for dimensions $N\geq 2$. The curved fronts concerned are transition fronts connecting $0$ and $1$. Under a priori assumption that there exist moving pulsating fronts in every direction, we show the existence of polytope-like curved fronts with $0$-zone being a polytope and $1$-zone being the complementary set. By reversing some conditions, we also show the existence of curved fronts with reversed $0$-zone and $1$-zone. Furthermore, the curved fronts constructed by us are proved to be unique and asymptotic stable.