Optimal critical mass in the two dimensional Keller–Segel model in $ {\mathbb{R}}^{2}$

Jean Dolbeault; Benoît Perthame

doi:10.1016/j.crma.2004.08.011

Partial Differential Equations

Optimal critical mass in the two dimensional Keller–Segel model in

R^{2}

[Masse critique optimale pour le modèle de Keller–Segel dans

R^{2}

Présenté par : Pierre-Louis Lions

Jean Dolbeault ¹ ; Benoît Perthame ²

¹ Ceremade (UMR CNRS no. 7534), université Paris IX-Dauphine, place de Lattre de Tassigny, 75775 Paris cedex 16, France
² DMA (UMR CNRS no. 8553), École normale supérieure, 45, rue d'Ulm, 75005 Paris cedex 05, France

Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 611-616.

Résumé
Abstract

Le système de Keller–Segel décrit le mouvement collectif de cellules attirées par une substance chimique et qui sont capables de l'émettre. Dans sa forme la plus simple, il s'agit d'une équation de dérive-diffusion pour la densité de cellules, couplée à une équation elliptique pour la concentration de chémo-attracteur. Il est bien connu qu'en deux dimensions, il y a existence pour des masses petites et explosion pour des masses grandes. Dans cette Note nous complétons ce résultat en donnant une expression de la masse critique dans le cas où le problème estposé dans tout l'espace.

The Keller–Segel system describes the collective motion of cells that are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. It is known that, in two space dimensions, for small initial mass there is global existence of classical solutions and for large initial mass blow-up occurs. In this Note we complete this picture and give an explicit value for the critical mass when the system is set in the whole space.

Reçu le : 2004-06-17
Accepté le : 2004-08-28
Publié le : 2004-10-14

DOI : 10.1016/j.crma.2004.08.011

Affiliations des auteurs :

Jean Dolbeault ¹ ; Benoît Perthame ²

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     title = {Optimal critical mass in the two dimensional {Keller{\textendash}Segel} model in $ {\mathbb{R}}^{2}$},
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Jean Dolbeault; Benoît Perthame. Optimal critical mass in the two dimensional Keller–Segel model in $ {\mathbb{R}}^{2}$. Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 611-616. doi : 10.1016/j.crma.2004.08.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.08.011/

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