%0 Journal Article
%T Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions
%A Adrien Blanchet
%A Jean Dolbeault
%A Benoit Perthame
%J Electronic Journal of Differential Equations
%D 2006
%I Texas State University
%X 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.
%K Keller-Segel model
%K existence
%K weak solutions
%K free energy
%K entropy method
%K logarithmic Hardy-Littlewood-Sobolev inequality
%K critical mass
%K Aubin-Lions compactness method
%K hypercontractivity
%K large time behavior
%K time-dependent rescaling
%K self-similar variables
%K intermediate asymptotics.
%U http://ejde.math.txstate.edu/Volumes/2006/44/abstr.html