The leading edge of a free boundary interacting with a line of fast diusion

TL;DR

This paper investigates the behavior of expanding fronts in a reaction-diffusion system with a line of fast diffusion, explaining an unexpected numerical observation through the construction and analysis of traveling wave solutions.

Contribution

It introduces a free boundary model related to biological invasion systems and rigorously explains the front's position using traveling wave analysis.

Findings

The leading edge is located in the lower half plane, not on the line.

Traveling wave solutions match numerical simulations.

The free boundary near the line exhibits the predicted behavior.

Abstract

The goal of this work is to explain an unexpected feature of the expanding level sets of the solutions of a system where a half plane, in which reaction-diusion phenomena take place, exchanges mass with a line having a large diusion of its own. The system was proposed by H. Berestycki, L. Rossi and the second author [4] as a model of enhancement of biological invasions by a line of fast diusion. It was observed numerically by A.-C. Coulon [7] that the leading edge of the front, rather than being located on the line, was in the lower half plane. We explain this behaviour for a free boundary problem much related to the system for which the simulations were made. We construct travelling waves for this problem, analyse their free boundary near the line, and prove that it has the behaviour predicted by the numerical simulations.