ac{\eta_K}{n(\gamma-1)}g_{ij}\right|^2vu\,dV\\
&-2\sigma_K\beta_K\int_M\left[(\gamma-1)({\rm Ric}+Kg)(\nabla v,\nabla v )+((\gamma-1)\Delta v+\eta_K)^2\right]vu\,dV,\notag\end{aligned}$$ where $a=\frac{n(\gamma-1)}{n(\gamma-1)+2}$, $\kappa=K\sup\limits_{M\times(0,T]}u^{\gamma-1}$, $\sigma_K=\left(\frac{e^{2{\kappa}t}-1}{2\kappa}\right)^{a}$, $\beta_K=\frac{\sinh(2\kappa t)}{2\kappa}$ and $\eta_K=\frac{2a\kappa}{1-e^{-2\kappa t}}$. Moreover, if ${\rm Ric}\ge-Kg$ for $K\ge0$, then $\mathcal{W}_{K}(v,t)$ is monotone decreasing along the porous medium equation .

When $K=0$, $\sigma_0=t^{a}$, $\beta_0=t$ and $\eta_0=\frac{a}t$, the entropy formula in Theorem \[KPMEentropy\] reduced the result of Lu-Ni-Vazquez-Villani in [@LNVV], $$\begin{aligned}
\label{PMEentropy2}
\frac{d}{dt}\mathcal{W}_0(v,t)=&-2(\gamma-1)t^{a+1}\int_M\left[\Big|\nabla_i\nabla_jv
+\frac{a}{n(\gamma-1)t}g_{ij}\Big|^2+ {\rm Ric}(\nabla v,\nabla v)\right]vu\,dV\notag\\
&-2t^{a+1}\int_M\left[(\gamma-1)\Delta v+\frac {a}t\right]^2 vu\,dV,\end{aligned}$$ where $$\label{PMEentropy1}
\mathcal{W}_0(v,t)= t^{a+1}\int_M\left(\gamma\frac{|\nabla v|^2}{v}-\frac{a+1}t\right)vu\,dV.$$

A weighted Riemannian manifold is a Riemannian manifold $(M,g)$ with a smooth measure $d\mu:= e^{-f}\,{dV}$, denoted by $(M,g, d\mu)$, where $f$ is a smooth function on $M$. The weighted Riemannian manifold carries a natural analog of the Ricci curvature, that is, the $m$-Bakry-Émery Ricci curvature, which is defined as $${\rm Ric}_f^m:={\rm Ric}+\nabla\nabla f-\frac{\nabla f\otimes \nabla f}{m-n},\quad (n\leq m\leq \infty).$$ In particular, when $m=\infty$, ${\rm Ric}^{\infty}_f={\rm Ric}_f:={\rm Ric}+\nabla\nabla f$ is the classical Bakry–Émery Ricci curvature, which was introduced in the study of diffusion processes and functional inequalities including the Poincaré and the logarithmic Sobolev inequalities(see [@BGL] for a comprehensive introduction), then it is extensively investigated in the theory of the Ricci flow (for example, the gradient shrinking Ricci soliton equ