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

Ali Taheri; Vahideh Vahidifar

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arXiv · 2303.05802

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

Ali Taheri·Vahideh Vahidifar

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Abstract

This article presents new parabolic and elliptic type gradient estimates for positive smooth solutions to a nonlinear parabolic equation 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-\'Emery 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.

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Ali Taheri, Vahideh Vahidifar. 2023-03-10. Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials. https://arxiv.org/abs/2303.05802

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