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

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 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.

💡 Research Summary

The paper investigates the parabolic‑elliptic Keller‑Segel system in ℝⁿ with the critical diffusion exponent m = 2 − 2/d, a regime where the nonlinear diffusion Δρᵐ and the chemotactic drift −∇·(ρ∇u) balance each other. By introducing the pressure variable p (log ρ for d = 2, and (m/(m‑1)) ρ^{m‑1} for d > 2) and the combined potential v = p − u, the authors rewrite the system as ∂ₜρ = ∇·(ρ∇v). Their main achievement is to prove Li‑Yau‑type lower bounds for Δv in two dimensions and Aronson‑Bénilan‑type lower bounds for Δv in higher dimensions, despite the presence of the non‑local chemotactic term u.

The results are organized into three mass regimes: (i) small mass (M < ε_d, with ε_d explicitly computable), (ii) sub‑critical mass (0 < M < M_c, where M_c(2)=8π and M_c(d>2)=2 C* (m‑1)^{1/(1‑m)}), and (iii) critical mass (M = M_c). For each case the authors obtain a differential inequality of the form
 Δv ≥ −C t⁻¹ or Δv ≥ −C(T)/(1 + t),
with constants C depending only on the total mass (and, in the sub‑critical case, on the initial free energy). These inequalities immediately yield uniform L^∞‑bounds on the density: ∥ρ(t)∥∞ ≤ C t⁻¹ (small mass) or ∥ρ(t)∥∞ ≤ C(T)/(1 + t) (sub‑critical and critical cases).

A crucial technical component is the analysis of subsolutions to the elliptic inequalities Δlog ρ + ρ ≥ 0 (in d = 2) and Δh + c h^q ≥ 0 (in d > 2, with q = d/(d‑2)). The authors prove that the minimal mass for such subsolutions coincides with the classical critical mass in dimensions two and three, and they provide strong numerical evidence for the same equality in higher dimensions. This connection allows them to replace the usual second‑moment condition by a weaker logarithmic moment condition, which is natural for Keller‑Segel dynamics.

Using the Hardy–Littlewood–Sobolev (HLS) inequality (and its logarithmic counterpart in two dimensions), they relate the free energy functional
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