An interpolating Harnack inequality for nonlinear heat equation on a surface
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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*}
Abstract
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*}
참고문헌 (14)
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