Lipschitz property of bistable or combustion fronts and its applications

TL;DR

This paper investigates the Lipschitz regularity of level sets in reaction-diffusion equations, establishing conditions under which these sets are Lipschitz graphs and analyzing their large-scale behavior and wave speeds.

Contribution

It proves Lipschitz regularity of level sets for solutions of reaction-diffusion equations and characterizes the minimal wave speed using blowing down analysis.

Findings

Level sets are Lipschitz graphs near certain levels.

All level sets are Lipschitz if the solution connects 0 and 1.

Large scale motion law and minimal wave speed are characterized.

Abstract

For a class of reaction-diffusion equations describing propagation phenomena, we prove that for any entire solution $u$, the level set $\{u=\lambda\}$ is a Lipschitz graph in the time direction if $\lambda$ is close to $1$. Under a further assumption that $u$ connects $0$ and $1$, it is shown that all level sets are Lipschitz graphs. By a blowing down analysis, the large scale motion law for these level sets and a characterization of the minimal speed for travelling waves are also given.