Existence of travelling waves for a reaction-diffusion system with a line of fast diffusion

TL;DR

This paper proves the existence and uniqueness of travelling wave solutions in a coupled reaction-diffusion system involving a line of fast diffusion, using a novel continuation method and singular perturbation analysis.

Contribution

It introduces a new approach combining continuation methods and singular perturbation techniques to analyze travelling waves in coupled reaction-diffusion systems.

Findings

Existence and uniqueness of travelling waves established.

Connection to Wentzell boundary value problems through singular perturbation.

New analytical framework for systems with fast diffusion lines.

Abstract

We prove existence and uniqueness of travelling waves for a reaction-diffusion system coupling a classical reaction-diffusion equation in a strip with a diffusion equation on a line. To do this we use a continuation method which leads to further insight into the system. In particular, the transition occurs through a singular perturbation which seems new in this context, connecting the system with a Wentzell type boundary value problem.