[Masse critique optimale pour le modèle de Keller–Segel dans .]
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.
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@article{CRMATH_2004__339_9_611_0,
author = {Dolbeault, Jean and Perthame, Beno{\^\i}t},
title = {Optimal critical mass in the two dimensional {Keller{\textendash}Segel} model in $ {\mathbb{R}}^{2}$},
journal = {Comptes Rendus. Math\'ematique},
pages = {611--616},
publisher = {Elsevier},
volume = {339},
number = {9},
year = {2004},
doi = {10.1016/j.crma.2004.08.011},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2004.08.011/}
}
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AU - Dolbeault, Jean
AU - Perthame, Benoît
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JO - Comptes Rendus. Mathématique
PY - 2004
SP - 611
EP - 616
VL - 339
IS - 9
PB - Elsevier
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DO - 10.1016/j.crma.2004.08.011
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ID - CRMATH_2004__339_9_611_0
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Dolbeault, Jean; Perthame, Benoît. Optimal critical mass in the two dimensional Keller–Segel model in $ {\mathbb{R}}^{2}$. Comptes Rendus. Mathématique, Tome 339 (2004) no. 9, pp. 611-616. doi : 10.1016/j.crma.2004.08.011. http://www.numdam.org/articles/10.1016/j.crma.2004.08.011/
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