Upper bounds for the blow-up time of the 2-D parabolic-elliptic Patlak-Keller-Segel model of chemotaxis

In this paper, we obtain upper bounds for the critical time $T^{\ast }$ of the blow-up for the parabolic-elliptic Patlak-Keller-Segel system on the 2D-Euclidean space. No moment condition or/and entropy condition are required on the initial data; only the usual assumptions of non-negativity and finiteness of the mass is assumed. The result is expressed not only in terms of the supercritical mass $M>8\pi$, but also in terms of the {\it shape} of the initial data.