The global harnack estimates for a nonlinear heat equation with potential under finsler-geometric flow

Abstract

Abstract Let (Mn , F(t), m), t ∈ [0, T], be a compact Finsler manifold with F(t) evolving by the Finsler-geometric flow $\begin{array}{} \displaystyle \frac{\partial g(x,t)}{\partial t}=2h(x,t), \end{array}$ where g(t) is the symmetric metric tensor associated with F, and h(t) is a symmetric (0, 2)-tensor. In this paper, we consider local Li-Yau type gradient estimates for positive solutions of the following nonlinear heat equation with potential $$\begin{array}{} \displaystyle \partial_{t}u(x,t)=\Delta_{m}u(x,t)-\mathcal{R}(x,t)u(x,t) -au(x,t)\log u(x,t),\quad(x,t)\in M\times [0,T], \end{array}$$ along the Finsler-geometric flow, where 𝓡 is a smooth function, and a is a real nonpositive constant. As an application we obtain a global estimate and a Harnack estimate. Our results are also natural extension of similar results on Riemannian-geometric flow.