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Algebra i Analiz, 2020, Volume 32, Issue 3, Pages 149–179 (Mi aa1703)  
Research Papers

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

L. A. Caffarellia, J.-M. Roquejoffreb

a The University of Texas at Austin, Mathematics Department RLM 8.100, 2515 Speedway Stop C1200, Austin, Texas 78712-1202, U.S.A.
b Institut de Mathématiques de Toulouse (UMR CNRS 5219), Université Toulouse III, 118 route de Narbonne, 31062 Toulouse cedex, France
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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-diffusion phenomena take place exchanges mass with a line having a large diffusion of its own. The system was proposed by H. Berestycki, L. Rossi and the second author as a model of enhancement of biological invasions by a line of fast diffusion. It was observed numerically by A.-C. Coulon that the leading edge of the front, rather than being located on the line, was in the lower half-plane.
We explain this behavior for a closely related free boundary problem. We construct travelling waves for this problem, and the analysis of their free boundary near the line confirms the predictions of the numerical simulations.
Keywords: expanding level sets, reaction-diffusion phenomena, line of fast diffusion.
Funding agency Grant number
National Science Foundation DMS-1160802
European Union's Seventh Framework Programme 321186
L. A. Caffarelli is supported by NSF grant DMS-1160802. The research of J.-M. Roquejoffre has received funding from the ERC under the European Union's Seventh Frame work Programme (FP/2007-2013) / ERC Grant Agreement 321186 — ReaDi.
Received: 29.07.2019
English version:
St. Petersburg Mathematical Journal, 2021, Volume 32, Issue 3, Pages 499–522
DOI: https://doi.org/10.1090/spmj/1658
Document Type: Article
Language: English
Citation: L. A. Caffarelli, J.-M. Roquejoffre, “The leading edge of a free boundary interacting with a line of fast diffusion”, Algebra i Analiz, 32:3 (2020), 149–179; St. Petersburg Math. J., 32:3 (2021), 499–522
Citation in format AMSBIB
\Bibitem{CafRoq20}
\by L.~A.~Caffarelli, J.-M.~Roquejoffre
\paper The leading edge of a free boundary interacting with a line of fast diffusion
\jour Algebra i Analiz
\yr 2020
\vol 32
\issue 3
\pages 149--179
\mathnet{http://mi.mathnet.ru/aa1703}
\transl
\jour St. Petersburg Math. J.
\yr 2021
\vol 32
\issue 3
\pages 499--522
\crossref{https://doi.org/10.1090/spmj/1658}
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    		Алгебра и анализ St. Petersburg Mathematical Journal
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