Harnack estimates for the porous medium equation with potential under geometric flow

Abstract

Let $(M,g(t))$, $t\in [0,T)$ be a closed Riemannian $n$-manifold whose Riemannian metric $g(t)$ evolves by the geometric flow $\frac{\partial }{\partial t}{g}_{ij}=-2S_{ij}$, where $S_{ij}(t)$ is a symmetric two-tensor on $(M,g(t))$. We discuss differential Harnack estimates for positive solution to the porous medium equation with potential, $\frac{\partial u}{\partial t}=\Delta u^p+Su$, where $S=g^{ij}{S}_{ij}$ is the trace of $S_{ij}$, on time-dependent Riemannian metric evolving by the above geometric flow.