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Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics$g_{ij}(t)$ evolving by geometric flow $\partial_{t}g_{ij} = -2{S}_{ij}$, where$S_{ij}(t)$ is a family of smooth symmetric two-tensors on $M$. In this paperwe derive differential Harnack estimates for positive solutions to thenonlinear heat equation with potential: \begin{eqnarray*} \frac{\partialf}{\partial t} = {\Delta}f + \gamma (t) f\log f +aSf, \end{eqnarray*} where$\gamma (t)$ is a continuous function on $t$, $a$ is a constant and$S=g^{ij}S_{ij}$ is the trace of $S_{ij}$. Our Harnack estimates include manyknown results as special cases, and moreover lead to new Harnack inequalitiesfor a variety geometric flows.

Harnack Estimates for Nonlinear Heat Equations with Potentials in Geometric Flows