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arXiv · 2512.17772

A Li-Yau and Aronson-B\'enilan approach for the Keller-Segel system with critical exponent

Charles Elbar·Alejandro Fernández-Jiménez·Filippo Santambrogio
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Abstract

We prove Li-Yau and Aronson-B\'enilan 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 the critical mass case for $d \in \{ 2, 3 \}$. 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. Our argument requires a careful study of the subsolutions of the Liouville and Lane-Emden equations arising in the model.

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Charles Elbar, Alejandro Fernández-Jiménez, Filippo Santambrogio. 2025-12-19. A Li-Yau and Aronson-B\'enilan approach for the Keller-Segel system with critical exponent. https://arxiv.org/abs/2512.17772

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