Convergence to a terrace solution in multistable reaction-diffusion equations with discontinuities

TL;DR

This paper investigates the long-term behavior of solutions to reaction-diffusion equations with discontinuities, demonstrating convergence to traveling waves or terraces depending on the multistability of the system.

Contribution

It establishes the convergence of solutions to either traveling waves or terraces in multistable reaction-diffusion equations with discontinuities.

Findings

Solutions converge to traveling waves in bistable cases

Solutions form terraces in multistable cases

Convergence occurs despite discontinuities in the system

Abstract

In this paper we address the large-time behavior of solutions of bistable and multistable reaction-diffusion equations with discontinuities around the stable steady states. We show that the solution always converges to a special solution, which may either be a traveling wave in the bistable case, or more generally a terrace (i.e. a collection of stacked traveling waves with ordered speeds) in the multistable case.