Lipschitz property of bistable or combustion fronts and its applications

Abstract

For a class of reaction-diffusion equations describing propagation phenomena, we prove that for any entire solution u, the level set {u=λ} is a Lipschitz graph in the time direction if λ 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 traveling waves are also given.