Gradient Estimates for a Nonlinear Heat Equation Under Finsler-Geometric Flow

Abstract

This paper considers a compact Finsler manifold (M n ,F(t),m) evolving under a Finsler-geometric flow and establishes global gradient estimates for positive solutions of the following nonlinear heat equation ∂ t u(x,t) = ∆ m u(x,t), (x,t) ∈ M×[0,T], where ∆ m is the Finsler-Laplacian. By integrating the gradient estimates, we derive the corresponding Harnack inequalities. Our results generalize and correct the work of S. Lakzian, who established similar results for the Finsler-Ricci flow. Our results are also natural extension of similar results on Riemannian-geometric flow, previously studied by J. Sun. Finally, we give an application to the Finsler-Yamabe flow.