preprint

Inserted: 22 dec 2025
Last Updated: 6 oct 2026

Year: 2025

Inthis new version the equality between the critical mass defined in terms of subsolutions and that defined in terms of the HLS inequality has been extended to every dimension.

Abstract:

We prove Li-Yau and Aronson-Bénilan type estimates for the parabolic-elliptic Keller-Segel system with critical exponent $m=2-\frac 2d$, i.e. lower bounds on the Laplacian of a suitable notion of pressure in any dimension. We show that these estimates entail $L^{\infty}$ bounds on the density, depending on its initial mass, up to a certain critical mass. The relevant critical mass here is the minimal mass of subsolutions of a certain Liouville or Lane-Emden equation that we have to carefully study, but we prove that this coincides with the standard critical mass in the Keller-Segel system coming from HLS or log-HLS inequalities. We deduce from these results the global existence of smooth solutions in two cases: first, when the initial data is merely a measure but has sufficiently small mass; and second, when the initial free energy is bounded, and the mass is subcritical or critical.

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