Existence of multi-dimensional pulsating fronts for KPP equations: a new formulation approach

TL;DR

This paper offers a new formulation approach to prove the existence of multi-dimensional pulsating fronts in KPP equations, simplifying the analysis and revealing periodicity in the wave profiles along rational directions.

Contribution

It provides an alternative proof for pulsating front existence in multi-dimensional periodic media using a wave profile construction in a moving frame.

Findings

Existence of pulsating travelling fronts in multi-dimensional periodic KPP equations.

Wave profiles are periodic in time along rational directions.

The new proof simplifies previous technical challenges.

Abstract

This paper is concerned with the existence of pulsating travelling fronts for a KPP reaction-diffusion equation posed in a multi-dimensional periodic medium. We provide an alternative proof of the classic existence result. Our proof relies largely on the construction of a wave profile under a moving frame, which avoids many technical difficulties in dealing with degenerate elliptic equations. Intriguingly, our analysis also yields that the profile of the front propagating along each rational direction in $\mathbb{S}^{N-1}$ is periodic in time.