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

Abstract

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.

arXiv commentThe irrelevant discussion of the second moment condition has been removed in the shorter version published in J. Math. Anal. Appl. 549 (2025), no. 2, article no. 129487

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