Affiliation:
1. Department of pure Mathematics , Faculty of Science Imam Khomeini International University , Qazvin , Iran
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.
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