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

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 partial derivative(t)u(x,t) = Delta(m)u(x,t), (x,t) is an element of Mx[0,T], where Delta(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.