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

This article presents new gradient estimates for positive solutions to the nonlinear fast diffusion equation on smooth metric measure spaces, involving the <i>f</i>-Laplacian. The gradient estimates of interest are of Hamilton-Souplet-Zhang or elliptic type and are established using different 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-Émery <i>m</i>-Ricci curvature tensor. The curious 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.