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Publication 146

Front propagation directed by a line of fast diffusion: large diffusion and large time asymptotics


CMULaurent Dietrich
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

Jean-Michel Roquejoffre
Institut de Mathématiques de Toulouse
Université Paul Sabatier
118 route de Narbonne, F-31062 Toulouse Cedex 4, France

The system under study is a reaction-diffusion equation in a horizontal strip, coupled to a diffusion equation on its upper boundary via an exchange condition of the Robin type. This class of models was introduced by H. Berestycki, L. Rossi and the second author in order to model biological invasions directed by lines of fast diffusion. They proved, in particular, that the speed of invasion was enhanced by a fast diffusion on the line, the spreading velocity being asymptotically proportional to the square root of the fast diffusion coefficient. These results could be reduced, in the logistic case, to explicit algebraic computations. The goal of this paper is to prove that the same phenomenon holds, with a different type of nonlinearity, which precludes explicit computations. We discover a new transition phenomenon, that we explain in detail.
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