Front propagation directed by a line of fast diffusion: large diffusion and large time asymptotics
[Propagation de fronts dirigée par une ligne de diffusion rapide : limite en temps long et grande diffusion]
Dietrich, Laurent1 ; Roquejoffre, Jean-Michel2
1 Department of Mathematical Sciences, Carnegie Mellon University Pittsburgh, PA 15213, U.S.A.
2 Institut de Mathématiques de Toulouse, Université Paul Sabatier 118 route de Narbonne, F-31062 Toulouse Cedex 4, France
Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 141-176.

Nous étudions une équation de réaction-diffusion posée dans une bande horizontale, couplée à une équation de diffusion sur son bord supérieur à travers une condition de Robin. Cette classe de modèles a été proposée par H. Berestycki, L. Rossi et le deuxième auteur afin d’étudier l’influence d’une ligne de diffusion rapide (par exemple une route) sur les invasions biologiques. Ils prouvent que la vitesse d’invasion est augmentée par une forte diffusivité sur la ligne, et plus précisément asymptotiquement proportionnelle à la racine carrée de cette dernière. Dans le cas d’une croissance logistique, ces résultats peuvent être réduits à des calculs algébriques. Le but de cet article est de généraliser ce résultat à des non-linéarités différentes et pour lesquelles ces calculs ne peuvent être accomplis. Nous mettons aussi en lumière un nouveau phénomène de transition entre deux ondes progressives différentes, qu’on explique en détail.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.40
Classification : 35K57, 35B40, 35C07
Keywords: Reaction-diffusion, traveling fronts, asymptotic, enhancement, acceleration, propagation, coupling, invasion, quenching
Mot clés : Réaction-diffusion, fronts, ondes progressives, asymptotique, accélération, propagation, couplage, invasion, extinction
Dietrich, Laurent 1 ; Roquejoffre, Jean-Michel 2

1 Department of Mathematical Sciences, Carnegie Mellon University Pittsburgh, PA 15213, U.S.A.
2 Institut de Mathématiques de Toulouse, Université Paul Sabatier 118 route de Narbonne, F-31062 Toulouse Cedex 4, France 
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Dietrich, Laurent; Roquejoffre, Jean-Michel. Front propagation directed by a line of fast diffusion: large diffusion and large time asymptotics. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 141-176. doi : 10.5802/jep.40. http://www.numdam.org/articles/10.5802/jep.40/

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