The nonlinear porous medium equation for the f-Laplacian: Hamilton-Souplet-Zhang type gradient estimates and implications

TL;DR

This paper derives new Hamilton-Souplet-Zhang type gradient estimates for positive solutions of the nonlinear porous medium equation involving the f-Laplacian on smooth metric measure spaces, with implications for Liouville theorems and ancient solutions.

Contribution

It introduces novel gradient estimates for the NPME with the f-Laplacian in evolving metric measure spaces, extending previous results and providing new applications.

Findings

Established gradient estimates for NPME solutions

Derived implications for Liouville-type theorems

Characterized ancient solutions in this setting

Abstract

This article presents new gradient estimates for positive solutions to the nonlinear porous medium equation (NPME) in the context of smooth metric measure spaces. The diffusion operator here is the f-Laplacian and the gradient estimates of interest are mainly of Hamilton-Souplet-Zhang types. These estimates are established using a variety of methods and techniques and several implications, most notably, to parabolic Liouville-type results and characterisation of ancient solutions are given. The problem is posed in the general framework where the metric and potential evolve with time and the proofs make use of natural lower bounds on the time derivative of the metric and the Bakry-\'Emery m-Ricci curvature tensors. Our results extend and improve various existing ones in the literature.