Gradient Estimates and Harnack Inequality for Positive Solutions to the Heat Equation Under Compact Finsler Geometric Flows

Abstract

In this manuscript, we establish global Li–Yau-type gradient estimates for positive solutions to the heat equation associated with the time-dependent nonlinear Finsler Laplacian under a compact Finsler geometric flow. The estimates rely on the curvature-dimension condition and suitable bounds on the flow tensor. As a direct application, we derive the corresponding Harnack inequality. A key point is an operator exchange formula that accounts for the contribution of the reference measure.

Keywords

Finsler geometric flow / gradient estimates / heat flow / curvature-dimension condition / Harnack inequality / 53C60 / 58J35

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