A Li-Yau and Aronson-Bénilan approach for the Keller-Segel system with critical exponent

Abstract

We prove Li-Yau and Aronson-Bénilan type estimates for the parabolic-elliptic Keller-Segel system with critical exponent $m=2-\frac{2}{d}$, 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.

65 pages + Appendix + references