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

TLDR: 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 coupled to an elliptic equation for the chemo-attractant concentration as mentioned in this paper.

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