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

Lucilla Corrias; Benoît Perthame

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

Lucilla Corrias ¹ ; Benoît Perthame 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, Volume 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 :

Lucilla Corrias ¹ ; Benoît Perthame ²

@article{CRMATH_2006__342_10_745_0,
     author = {Lucilla Corrias and Beno{\^\i}t Perthame},
     title = {Critical space for the parabolic-parabolic {Keller{\textendash}Segel} model in $ {\mathbb{R}}^{d}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {745--750},
     publisher = {Elsevier},
     volume = {342},
     number = {10},
     year = {2006},
     doi = {10.1016/j.crma.2006.03.008},
     language = {en},
}

TY  - JOUR
AU  - Lucilla Corrias
AU  - Benoît Perthame
TI  - Critical space for the parabolic-parabolic Keller–Segel model in $ {\mathbb{R}}^{d}$
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 745
EP  - 750
VL  - 342
IS  - 10
PB  - Elsevier
DO  - 10.1016/j.crma.2006.03.008
LA  - en
ID  - CRMATH_2006__342_10_745_0
ER  -

%0 Journal Article
%A Lucilla Corrias
%A Benoît Perthame
%T Critical space for the parabolic-parabolic Keller–Segel model in $ {\mathbb{R}}^{d}$
%J Comptes Rendus. Mathématique
%D 2006
%P 745-750
%V 342
%N 10
%I Elsevier
%R 10.1016/j.crma.2006.03.008
%G en
%F CRMATH_2006__342_10_745_0

Lucilla Corrias; Benoît Perthame. Critical space for the parabolic-parabolic Keller–Segel model in $ {\mathbb{R}}^{d}$. Comptes Rendus. Mathématique, Volume 342 (2006) no. 10, pp. 745-750. doi : 10.1016/j.crma.2006.03.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.03.008/

Bibliographie
Cité par

[1] P. Biler, G. Karch, P. Laurençot, The 8π-problem for radially symmetric solutions of a chemotaxis model in a disc, Topol. Methods Nonlin. Anal. (2005), in press

[2] A. Blanchet, J. Dolbeault, B. Perthame, Two-dimensional Keller–Segel model: Optimal critical mass and qualitative properties of the solutions, (2006), in preparation

[3] M.P. Brenner; P. Constantin; L.P. Kadanoff; A. Schenkel; S.C. Venkataramani Diffusion, attraction and collapse, Nonlinearity, Volume 12 (1999), pp. 1071-1098

[4] V. Calvez, B. Perthame, M. Sharifi Tabar, Modified Keller–Segel system and critical mass for the log interaction kernel, (2006), in preparation

[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] J. Dolbeault; B. Perthame Optimal critical mass in the two-dimensional Keller–Segel model in $R^{2}$ , C. R. Math. Acad. Sci. Paris, Ser. I, Volume 339 (2004), pp. 611-616

[7] M.A. Herrero; E. Medina; J.J.L. Velázquez Self-similar blow-up for a reaction–diffusion system, J. Comput. Appl. Math., Volume 97 (1998), pp. 99-119

[8] D. Horstmann From 1970 until now: The Keller–Segel model in chemotaxis and its consequences, Jahresber. Deutch. Math.-Verein., Volume 105 (2003), pp. 103-165

[9] D. Horstmann From 1970 until now: The Keller–Segel model in chemotaxis and its consequences, Jahresber. Deutch. Math.-Verein., 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), pp. 819-824

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

[12] J.D. Murray Spatial models and biomedical applications, Mathematical Biology, II, Interdiscip. Appl. Math., vol. 18, Springer-Verlag, New York, 2003

[13] 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), pp. 145-156

[14] T. Nagai; R. Syukuinn; M. Umesako Decay properties and asymptotic profiles of bounded solutions to a parabolic system of chemotaxis in $R^{n}$ , Funkcial. Ekvac., Volume 46 (2003), pp. 383-407

[15] M. Rascle; C. Ziti Finite time blow-up in some models of chemotaxis, J. Math. Biol., Volume 33 (1995), pp. 388-414

[16] J. Renclawowicz, T. Hillen, Analysis of an attraction–repulsion chemotaxis model, Preprint, 2006

[17] J.J.L. Velázquez Stability of some mechanisms of chemotactic aggregation, SIAM J. Appl. Math., Volume 62 (2002), pp. 1581-1633 (electronic)

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

A mathematical model of mast cell response to acupuncture needling

Yannick Deleuze

C. R. Math (2013)

A one-dimensional Keller–Segel equation with a drift issued from the boundary

Vincent Calvez; Nicolas Meunier; Raphael Voituriez

C. R. Math (2010)

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

Jean Dolbeault; Benoît Perthame

C. R. Math (2004)