Blow-up of solutions to the Patlak-Keller-Segel equation in dimension ν≥2

We prove a blow-up criterion for the solutions to the ν-dimensional Patlak-Keller-Segel equation in the whole space. The condition is new in dimension three and higher. In dimension two, it is exactly Dolbeault's and Perthame's blow-up condition, i.e., blow-up occurs, if the total mass exceeds 8π.