Electron. J. Diff. Eqns., Vol. 2006(2006), No. 44, pp. 1-33.

Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions

Adrien Blanchet, Jean Dolbeault, Benoit Perthame

The Keller-Segel system describes the collective motion of cells which 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 solutions and for large initial mass blow-up occurs. In this paper we complete this picture and give a detailed proof of the existence of weak solutions below the critical mass, above which any solution blows-up in finite time in the whole Euclidean space. Using hypercontractivity methods, we establish regularity results which allow us to prove an inequality relating the free energy and its time derivative. For a solution with sub-critical mass, this allows us to give for large times an ``intermediate asymptotics'' description of the vanishing. In self-similar coordinates, we actually prove a convergence result to a limiting self-similar solution which is not a simple reflect of the diffusion.

Submitted February 28, 2006. Published April 6, 2006.
Math Subject Classifications: 35B45, 35B30, 35D05, 35K15, 35B40, 35D10, 35K60.
Key Words: Keller-Segel model; existence; weak solutions; free energy; entropy method; logarithmic Hardy-Littlewood-Sobolev inequality; critical mass; Aubin-Lions compactness method; hypercontractivity; large time behavior; time-dependent rescaling; self-similar variables; intermediate asymptotics.

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Adrien Blanchet
CEREMADE (UMR CNRS no. 7534), Université Paris Dauphine
Place de Lattre de Tassigny, 75775 Paris Cédex 16, France
and CERMICS, ENPC, 6-8 avenue Blaise Pascal
Champs-sur-Marne, 77455 Marne-la-Vallée Cédex 2, France
email: blanchet@ceremade.dauphine.fr
Jean Dolbeault
CEREMADE (UMR CNRS no. 7534), Université Paris Dauphine
Place de Lattre de Tassigny, 75775 Paris Cédex 16, France
Phone: (33) 1 44 05 46 78, Fax: (33) 1 44 05 45 99
email: dolbeaul@ceremade.dauphine.fr
Benoît Perthame
DMA (UMR CNRS no. 8553), Ecole Normale Supérieure
45 rue d'Ulm, 75005 Paris Cédex 05, France
email: Benoit.Perthame@ens.fr

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