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

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.