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

• Main Authors: Adrien Blanchet, Jean Dolbeault, Benoit Perthame
• Format: Article
• Language: English
• Published: Texas State University 2006-04-01
• Series: Electronic Journal of Differential Equations
• Subjects: 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.
• Online Access: http://ejde.math.txstate.edu/Volumes/2006/44/abstr.html

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.