Gradient estimates for [formula omitted] on manifolds and some Liouville-type theorems

Abstract

In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: u t = Δ F ( u ) , with F ′ ( u ) > 0 , on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equation (PME): u t = Δ ( u p ) , p > 0 , and obtain localized Hamilton-type gradient estimates for FDE and PME in a larger range of p than that for Aronson–Bénilan estimate, Harnack inequalities and Cauchy problems in the literature. Applying the localized gradient estimates for FDE and PME, we prove some Liouville-type theorems for positive global solutions of FDE and PME on noncompact complete manifolds with nonnegative Ricci curvature, generalizing Yauʼs celebrated Liouville theorem for positive harmonic functions.