Bulletin of the
Korean Mathematical Society BKMS

In this short note we prove new differential Harnack inequalities interpolating those for the static surface and for the Ricci flow. In particular, for $0\le \varepsilon \le 1 $, $\alpha\geq 0 $, $\beta\geq 0 $, $\gamma\leq1$ and $u$ being a positive solution to \begin{equation*} \frac{\partial u}{\partial t}=\Delta u-\alpha u\log u+\varepsilon Ru+\beta u^\gamma \end{equation*} on closed surfaces under the flow $\frac{\partial}{\partial t}g_{ij}=-\varepsilon Rg_{ij}$ with $R>0,$ we prove that \begin{equation*} \frac{\partial}{\partial t}\log u-|\nabla \log u|^2+\alpha \log u-\beta u^{\gamma-1}+\frac{1}{t}=\Delta \log u+\varepsilon R+\frac{1}{t} \geq 0. \end{equation*}

Keywords: Ricci flow, Harnack estimate, nonlinear heat equation

MSC numbers: 53C44

Supported by: Research supported by Zhejiang Provincial Natural Science Foundation of China (Grant Number LY18A010022) and NSFC (Grant Number 11971355).