Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials

Abstract

This article presents new parabolic and elliptic type gradient estimates for positive smooth solutions to nonlinear parabolic equations involving the Witten Laplacian in the context of smooth metric measure spaces. The metric and potential here are time dependent and evolve under a super Perelman–Ricci flow. The estimates are derived under natural lower bounds on the associated generalised Bakry–Émery Ricci curvature tensors and are utilised in establishing fairly general local and global bounds, Harnack-type inequalities and Liouville-type global constancy theorems to mention a few. Other implications and consequences of the results are also discussed.