Interaction of a free boundary with a diffusion on a plane: analogy with the obstacle problem

TL;DR

This paper explores how a free boundary interacting with a diffusion process on a plane resembles the obstacle problem, extending the analogy to higher dimensions and providing insights into biological invasion models.

Contribution

It advances understanding of free boundary behavior in reaction-diffusion systems by establishing a deeper analogy with the obstacle problem in higher dimensions.

Findings

The free boundary exhibits properties similar to the obstacle problem.

Numerical simulations are explained through the free boundary-obstacle analogy.

The analogy extends to higher-dimensional spaces, enriching the theoretical framework.

Abstract

A horizontal $N$-dimensional plane, having a diffusion of its own, exchanges with the lower half space. There, a reaction-diffusion process, modelled by a free boundary problem, takes place. We wish to understand whether, and how, the free boundary meets the plane. The origin of this problem is a two-dimensional reaction diffusion model proposed some time ago by the second author, in collaboration with H. Berestycki and L. Rossi, to model how biological invasions can be enhanced by a line of fast diffusion. Some counter-intuitive numerical simulations of this model, due to A.-C. Coulon, have been explained by the first two authors by transforming the model into a free boundary interacting with a line, and a careful study of the free boundary. At this occasion, it was noticed that the free boundary very much like that of the obstacle problem. The goal of the paper is to explain how this analogy with the obstacle problem can be pushed further in higher space dimensions.