Global existence and uniqueness of solutions for one-dimensional reaction-interface systems

TL;DR

This paper establishes the existence and uniqueness of solutions for a one-dimensional reaction-interface system modeling wave propagation with annihilation in excitable media, including solutions beyond annihilation events.

Contribution

It introduces a framework for weak solutions in reaction-interface systems with intersecting interfaces, ensuring well-posedness of the problem.

Findings

Proves existence and uniqueness of solutions.

Develops a notion of weak solutions for interface intersection.

Ensures well-posedness under certain conditions.

Abstract

In this paper, we provide a mathematical framework in studying the wave propagation with the annihilation phenomenon in excitable media. We deal with the existence and uniqueness of solutions to a one-dimensional free boundary problem (called a reaction--interface system) arising from the singular limit of a FitzHugh--Nagumo type reaction--diffusion system. Because of the presence of the annihilation, interfaces may intersect each other. We introduce the notion of weak solutions to study the continuation of solutions beyond the annihilation time. Under suitable conditions, we show that the free boundary problem is well-posed.