Gradient Estimates and Harnack Inequalities of a Nonlinear Heat Equation for the Finsler-Laplacian

Let (M-n, F, m) be an n-dimensional compact Finsler manifold. In this paper, we study the nonlinear heat equation partial derivative(t)u = Delta(m)u on M-n x [0, T], where Delta(m) is the Finsler-Laplacian. We derive Li-Yau type gradient estimates for positive global solutions of this equation on static Finsler manifolds, as well as under the Finsler-Ricci flow. As corollaries, in both cases, the corresponding Harnack inequalities are also obtained.