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

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

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     author = {Jean Dolbeault and Beno{\^\i}t Perthame},
     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},
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     number = {9},
     year = {2004},
     doi = {10.1016/j.crma.2004.08.011},
     language = {en},
}

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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/

Bibliographie
Cité par

[1] W. Beckner Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality, Ann. Math. (2), Volume 138 (1993) no. 1, pp. 213-242

[2] P. Biler; T. Nadzieja A class of nonlocal parabolic problems occurring in statistical mechanics, Colloq. Math., Volume 66 (1993) no. 1, pp. 131-145

[3] E. Carlen; M. Loss Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on $S^{n}$ , Geom. Funct. Anal., Volume 2 (1992) no. 1, pp. 90-104

[4] L. Corrias; B. Perthame; H. Zaag A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris, Ser. I, Volume 336 (2003) no. 2, pp. 141-146

[5] L. Corrias; B. Perthame; H. Zaag Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., Volume 72 (2004), pp. 1-29

[6] H. Gajewski; K. Zacharias Global behaviour of a reaction–diffusion system modelling chemotaxis, Math. Nachr., Volume 195 (1998), pp. 77-114

[7] M.A. Herrero; E. Medina; J.J.L. Velázquez Finite-time aggregation into a single point in a reaction–diffusion system, Nonlinearity, Volume 10 (1997) no. 6, pp. 1739-1754

[8] D. Horstmann On the existence of radially symmetric blow-up solutions for the Keller–Segel model, J. Math. Biol., Volume 44 (2002) no. 5, pp. 463-478

[9] D. Horstmann From 1970 until now: the Keller–Segel model in chemotaxis and its consequences, Jahresber. Deutsch. Math.-Verei., Volume 106 (2004), pp. 51-69

[10] W. Jäger; S. Luckhaus On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., Volume 329 (1992) no. 2, pp. 819-824

[11] E.F. Keller; L.A. Segel Model for chemotaxis, J. Theor. Biol., Volume 30 (1971), pp. 225-234

[12] P.K. Maini Applications of mathematical modelling to biological pattern formation, (Sitges, 2000) (Lecture Notes in Phys.), Volume vol. 567, Springer, Berlin (2001), pp. 205-217

[13] A. Marrocco Numerical simulation of chemotactic bacteria aggregation via mixed finite elements, Math. Model. Numer. Anal. (M2AN), Volume 37 (2003) no. 4, pp. 617-630

[14] J.D. Murray Mathematical Biology. II, Spatial Models and Biomedical Applications, Interdisciplinary Appl. Math., vol. 18, Springer-Verlag, New York, 2003

[15] T. Nagai; T. Senba Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis, Adv. Math. Sci. Appl., Volume 8 (1998) no. 1, pp. 145-156

[16] C.S. Patlak Random walk with persistence and external bias, B. Math. Biophys., Volume 15 (1953), pp. 311-338

[17] T. Senba; T. Suzuki Weak solutions to a parabolic-elliptic system of chemotaxis, J. Func. Anal., Volume 47 (2001), pp. 17-51

[18] J.J.L. Velázquez Stability of some mechanisms of chemotactic aggregation, SIAM J. Appl. Math., Volume 62 (2002) no. 5, pp. 1581-1633 (electronic)

[19] M.I. Weinstein Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., Volume 87 (1982/83) no. 4, pp. 567-576

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