Gradient estimates for $u_t=\Delta F(u)$ on manifolds and some Liouville-type theorems

TL;DR

This paper establishes localized Hamilton-type gradient estimates for solutions to porous media and fast diffusion equations on manifolds, leading to new Liouville-type theorems that generalize classical results to broader parameter ranges.

Contribution

It introduces new localized gradient estimates for porous media and fast diffusion equations on manifolds, extending the applicable range beyond previous bounds and deriving Liouville-type theorems.

Findings

Localized gradient estimates for FDE and PME on manifolds.

Extended the range of p for gradient estimates beyond Aronson-Bénilan.

Proved Liouville-type theorems for positive solutions on noncompact manifolds.

Abstract

In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: $$u_t=\Delta F(u),$$ 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): $$u_t=\Delta (u^p),\qquad p>0,$$ and obtain localized Hamilton-type gradient estimates for FDE and PME in a larger range of $p$ than that for Aronson-B\'enilan 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 Yaus celebrated Liouville theorem for positive harmonic functions.