  In this paper, we obtain upper bounds for the critical time $T^*$ 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.
