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

Abstract : 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.
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Contributor : Jean Dolbeault <>
Submitted on : Saturday, March 25, 2006 - 2:51:27 PM
Last modification on : Wednesday, May 6, 2020 - 3:56:28 PM
Document(s) archivé(s) le : Monday, September 17, 2012 - 1:00:36 PM

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• HAL Id : hal-00021782, version 1

### Citation

Adrien Blanchet, Jean Dolbeault, Benoît Perthame. Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions. Electronic Journal of Differential Equations, Texas State University, Department of Mathematics, 2006, 44, 32 pp. ⟨hal-00021782⟩

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