Critical space for the parabolic-parabolic Keller–Segel model in $ {\mathbb{R}}^{d}$

Corrias, Lucilla; Perthame, Benoît

doi:10.1016/j.crma.2006.03.008

Partial Differential Equations

Critical space for the parabolic-parabolic Keller–Segel model in

R^{d}

[Espace critique pour le modèle de Keller–Segel parabolique-parabolique dans

R^{d}

]

Présenté par : Philippe G. Ciarlet

Corrias, Lucilla ¹ ; Perthame, Benoît ²

¹ Département de mathématiques, Université d'Évry Val d'Essonne, rue du pere Jarlan, 91025 Evry cedex, France
² DMA (UMR CNRS no. 8553), École normale supérieure, 45, rue d'Ulm, 75005 Paris cedex 05, France

Comptes Rendus. Mathématique, Tome 342 (2006) no. 10, pp. 745-750.

Résumé
Abstract

Nous considérons le système de Keller–Segel posé sur $R^{d}$ dans le cas d'une équation parabolique sur le chemoattractant. Nous démontrons que l'espace critique, comme dans le cas elliptique, est que la densité bactérienne initiale vérifie $n_{0} \in L^{a} (R^{d})$ , $a > d / 2$ , et que la concentration initiale de chémoattractant vérifie $\nabla c_{0} \in L^{d} (R^{d})$ . Dans ces espaces, une donnée initiale petite donne des solutions globales qui tendent vers 0 en temps grand comme l'équation de la chaleur ainsi que des effets régularisants de type hypercontractifs.

We study the Keller–Segel system in $R^{d}$ when the chemoattractant concentration is described by a parabolic equation. We prove that the critical space, with some similarity to the elliptic case, is that the initial bacteria density satisfies $n_{0} \in L^{a} (R^{d})$ , $a > d / 2$ , and that the chemoattractant concentration satisfies $\nabla c_{0} \in L^{d} (R^{d})$ . In these spaces, we prove that small initial data give rise to global solutions that vanish as the heat equation for large times and that exhibit a regularizing effect of hypercontractivity type.

Reçu le : 2006-02-21
Accepté le : 2006-03-03
Publié le : 2006-04-11

DOI : 10.1016/j.crma.2006.03.008

Affiliations des auteurs :

Corrias, Lucilla ¹ ; Perthame, Benoît ²

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Corrias, Lucilla; Perthame, Benoît. Critical space for the parabolic-parabolic Keller–Segel model in $ {\mathbb{R}}^{d}$. Comptes Rendus. Mathématique, Tome 342 (2006) no. 10, pp. 745-750. doi : 10.1016/j.crma.2006.03.008. http://www.numdam.org/articles/10.1016/j.crma.2006.03.008/

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