ARONSON–BÉNILAN GRADIENT ESTIMATES FOR POROUS MEDIUM EQUATIONS UNDER LOWER BOUNDS OF N-WEIGHTED RICCI CURVATURE WITH N < 0

Abstract

The Aronson–Bénilan gradient estimate for the porous medium equation has been studied as a counterpart to the Li–Yau gradient estimate for the heat equation. In this paper, we give the Aronson–Bénilan gradient estimates for the porous medium equation on weighted Riemannian manifolds under lower bounds of N-weighted Ricci curvature with 𝜀-range for some N < 0. This is a generalization of those estimates under constant lower N-weighted Ricci curvature bounds with N ∈ [n,∞).