Hamilton and Li–Yau type gradient estimates for a weighted nonlinear parabolic equation under a super Perelman–Ricci flow

Abstract

In this paper we derive elliptic and parabolic type gradient estimates for positive smooth solutions to a class of nonlinear parabolic equations on smooth metric measure spaces where the metric and potential are time dependent and evolve under a super Perelman–Ricci flow. A number of implications, notably, a parabolic Harnack inequality, a class of Hamilton type dimension-free gradient estimates and two general Liouville type theorems along with their consequences are discussed. Some examples and special cases are presented to illustrate the results.