Blow-Up of Solutions to the Patlak-Keller-Segel Equation in Dimension $\nu\geq2$

Abstract

We prove a blow-up criterion for the solutions to the $\nu$-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 total mass exceeds $8\pi$ .