Cahiers du CEREMADE 

Unité
Mixte de Recherche du C.N.R.S. N°7534 

Abstract : The KellerSegel 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 driftdiffusion equation for the cell density coupled to an elliptic equation for the chemoattractant concentration. It is known that, in two space dimensions, for small initial mass, there is global existence of solutions and for large initial mass blowup 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 blowsup in finite time. 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 subcritical mass, this allows us to give for large times an intermediate asymptotics description of the vanishing. In selfsimilar coordinates, we actually prove a convergence result to a limiting selfsimilar solution which is not a simple reflect of the diffusion. 





200617 

02032006 

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