Asymptotic properties and classification of bistable fronts with Lipschitz level sets

Abstract

In this paper we study solutions to reaction-diffusion equations in the bistable case, defined on the whole space in dimension $N$. The existence of solutions with cylindric symmetry is already known. Here we prove the uniqueness of these cylindric solutions whose level sets are curved Lipschitz graphs. Using a centre manifold-like argument, we also give the precise asymptotics of these level sets at infinity. In dimension 2, we classify all solutions under weak conditions at infinity. Finally, we also provide an alternative proof of the existence of these solutions in dimension 2, based on a continuation argument.