Revisiting the blow-up criterion and the maximal existence time for solutions of the parabolic-Elliptic keller-Segel system in 2D-euclidean space

Abstract

In this paper, we revisit the blow-up criteria for the simplest parabolic-elliptic Patlak-Keller-Segel (PKS) system in the 2D Euclidean space, including a consumption term. In the supercritical mass case M > 8π, and under an additional global assumption on the second moment (or variance) of the initial data, we establish blow-up results for a broader class of initial conditions than those traditionally considered. We also derive improved upper bounds for the maximal existence time of (PKS) solutions on the plane. These time estimates are obtained through a sharp analysis of a one-parameter differential inequality governing the evolution of the second moment of the (PKS) system.

In particular, we obtain that for any n₀ with finite second order moment, λn₀ for λ sufficiently large provide an initial datum yielding blow up. The current blow up criterion is also compared to the available ones in the literature.