Compactly supported travelling waves arising in a general reaction-diffusion Stefan model

TL;DR

This paper investigates a reaction-diffusion Stefan model and discovers a new class of compactly supported travelling waves that exist over a range of speeds, with conditions for their existence and asymptotic solutions verified numerically.

Contribution

It introduces a general reaction-diffusion Stefan model with novel compactly supported travelling waves over a range of speeds, unlike traditional models.

Findings

Compactly supported travelling waves exist over a range of speeds.

Necessary conditions for existence of these waves are identified.

Asymptotic analysis matches numerical simulations accurately.

Abstract

We examine travelling wave solutions of the reaction-diffusion equation, $\partial_t u= R(u) + \partial_x \left[D(u) \partial_x u\right]$, with a Stefan-like condition at the edge of the moving front. With only a few assumptions on $R(u)$ and $D(u)$, a variety of new compactly supported travelling waves arise in this Reaction-Diffusion Stefan model. While other reaction-diffusion models admit compactly supported travelling waves for a unique wavespeed, we show that compactly supported travelling waves in the Reaction-Diffusion Stefan model exist over a range of wavespeeds. Furthermore, we determine the necessary conditions on $R(u)$ and $D(u)$ for which compactly supported travelling waves exist for all wavespeeds. Using asymptotic analysis in various distinguished limits of the wavespeed, we obtain approximate solutions of these travelling waves, agreeing with numerical simulations with high accuracy.