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Let $(M, g(t))$, $t\in[0,T)$ be a closed Riemannian $n$-manifold whoseRiemannian 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 tothe porous medium equation with potential, $\frac{\partial u}{\partialt}=\Delta u^{p}+S u$, where $S=g^{ij}S_{ij}$ is the trace of $S_{ij}$, ontime-dependent Riemannian metric evolving by the above geometric flow.

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