Terrace solutions for non-Lipschitz multistable nonlinearities

TL;DR

This paper investigates the existence and uniqueness of terrace solutions in reaction-diffusion equations with non-Lipschitz, discontinuous reaction functions, extending the understanding of complex invasion phenomena involving multiple stable states.

Contribution

It introduces the concept of terrace solutions for reaction-diffusion equations with discontinuous nonlinearities, establishing their existence and uniqueness.

Findings

Existence of terrace solutions for non-Lipschitz reaction functions

Uniqueness of the terrace solutions under certain conditions

Extension of traveling wave theory to multistable, discontinuous cases

Abstract

Traveling wave solutions of reaction-diffusion equations are well-studied for Lipschitz continuous monostable and bistable reaction functions. These special solutions play a key role in mathematical biology and in particular in the study of ecological invasions. However, if there are more than two stable steady states, the invasion phenomenon may become more intricate and involve intermediate steps, which leads one to consider not a single but a collection of traveling waves with ordered speeds. In this paper we show that, if the reaction function is discontinuous at the stable steady states, then such a collection of traveling waves exists and even provides a special solution which we call a terrace solution. More precisely, we will address both the existence and uniqueness of the terrace solution.