Optimal critical mass in the two-dimensional Keller-Segel model in $\mathbb R\sp 2$

The Keller-Segel system describes the collective motion of cells that 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 classical solutions and for large initial mass blow-up occurs. In this note we complete this picture and give an explicit value for the critical mass when the system is set in the whole space