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

## TL;DR

This paper derives differential Harnack estimates for positive solutions to the porous medium equation with potential on a manifold undergoing a geometric flow, extending classical results to evolving geometries.

## Contribution

It establishes new Harnack inequalities for the porous medium equation with potential under general geometric flows on closed manifolds.

## Key findings

- Derived differential Harnack estimates for the equation
- Extended classical results to time-dependent metrics
- Applicable to a broad class of geometric flows

## 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, $\frac{\partial u}{\partial t}=\Delta u^{p}+S u$, where $S=g^{ij}S_{ij}$ is the trace of $S_{ij}$, on time-dependent Riemannian metric evolving by the above geometric flow.