Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces

2024/04/02 by Ali Taheri, Taheri, Ali, Vahideh Vahidifar +1 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2404.01749

openalex publication_date 2024/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we prove gradient estimates of both elliptic and parabolic types, specifically, of Souplet-Zhang, Hamilton and Li-Yau types for positive smooth solutions to a class of nonlinear parabolic equations involving the Witten or drifting Laplacian on smooth metric measure spaces. These estimates are established under various curvature conditions and lower bounds on the generalised Bakry-Émery Ricci tensor and find utility in proving elliptic and parabolic Harnack-type inequalities as well as general elliptic and parabolic Liouville-type and other global constancy results. Several applications and consequences are presented and discussed.

Cited by

• The nonlinear porous medium equation for the f-Laplacian: Hamilton-Souplet-Zhang type gradient estimates and implications
• Li-Yau Estimates and Harnack Inequalities for Nonlinear Slow Diffusion Equations on a Smooth Metric Measure Space

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