The nonlinear fast diffusion equation on smooth metric measure spaces: Hamilton-Souplet-Zhang estimates and a Ricci-Perelman super flow

TL;DR

This paper develops new gradient estimates for positive solutions to the nonlinear fast diffusion equation on smooth metric measure spaces, revealing deep connections between geometry, nonlinearity, and evolution, with implications for Liouville theorems and ancient solutions.

Contribution

It introduces novel Hamilton-Souplet-Zhang type gradient estimates for the fast diffusion equation on evolving metric measure spaces, extending previous results to a more general geometric and nonlinear setting.

Findings

Established gradient estimates involving the $f$-Laplacian.

Derived implications for Liouville type theorems.

Characterized ancient solutions under the super flow.

Abstract

This article presents new gradient estimates for positive solutions to the nonlinear fast diffusion equation on smooth metric measure spaces, involving the $f$-Laplacian. The gradient estimates of interest are mainly of Hamilton-Souplet-Zhang or elliptic type and are proved using different set of methods and techniques. Various implications notably to parabolic Liouville type results and characterisation of ancient solutions are given. The problem is considered in the general setting where the metric and potential evolve under a super flow involving the Bakry-\'Emery $m$-Ricci curvature tensor. The remarkable interplay between geometry, nonlinearity, and evolution -- and their intricate roles in the estimates and the maximum exponent range of fast diffusion -- is at the core of the investigation.