Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 44, pp. 1-33.

Title: Two-dimensional Keller-Segel model: Optimal critical mass
       and qualitative properties of the solutions
       
Authors: Adrien Blanchet (Univ. Paris Dauphine, Paris,France)
         Jean Dolbeault  (Univ. Paris Dauphine, Paris, France)
	 Benoit Perthame (Ecole Normale Superieure, Paris, France)


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.
  
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.