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.