ome Perelman-Ni type entropy formulae for the linear heat equation and proved their monotonicity.

A natural question is how to establish the entropy formula for the nonlinear equation on Riemannian manifolds. Kotschwar-Ni [@KoNi] and Lu-Ni-Vazquez-Villani [@LNVV] obtained the entropy monotonicity formula for the $p$-Laplaican heat equation and the porous medium equation on compact Riemannian manifolds with nonnegative Ricci curvature respectively. In [@WC2], the author proved the entropy monotonicity formula for positive solution to the doubly nonlinear diffusion equation on the closed Riemannian manifolds with nonnegative Ricci curvature.

The first step in the study of the $\mathcal{W}$-entropy on Riemannian manifolds with negative lower bound of Ricci curvature is to find the suitable quantity to define the $\mathcal{W}$-entropy. In [@LiLi2; @LiLi3], S. Li and X.-D. Li gave a new idea to introduce the correct quantity for the definition of the $\mathcal{W}$-entropy for the heat equation of the Witten Laplacian on Riemannian manifolds with lower bound of the infinity dimensional Bakry-Emery Ricci curvature. Motivated by their works, we can obtain the Perelman type $\mathcal{W}$-entropy monotonicity formula for the porous medium equation on closed Riemannian manifolds with Ricci curvature (or Bakry-Emery Ricci curvature) bounded below.

\[KPMEentropy\] Let $(M,g)$ be a closed Riemannian manifold with Ricci curvature bounded below by $-K(K\ge0)$. Suppose $u$ be a smooth positive solution to the porous medium equation $$\label{PME}
\partial_tu=\Delta u^{\gamma}$$ and $v=\frac{\gamma}{\gamma-1}u^{\gamma-1}$ the pressure function. For any $\gamma>1$, define the Perelman-type $\mathcal{W}$-entropy $$\label{WKentropy}
\mathcal{W}_K(v,t):=\sigma_K\beta_K\int_M\left[\gamma\frac{|\nabla v|^2}{v}-\left(\frac{1}{\beta_K}+\frac{\dot{\sigma}_K}{\sigma_K}\right)\right]vu\,dV,$$ Then we have $$\begin{aligned}
\label{WKDentropy}
\frac{d}{dt}\mathcal{W}_K(v,t)
\le&-2\sigma_K\beta_K\int_M (\gamma-1) \left|\nabla_i\nabla_jv+\fr