gma}_K^2}{\sigma_K^2}\right)\mathcal{N}_K\notag\\
=&-2(\gamma-1)\sigma_K\int_M\left[|\nabla\nabla v|^2+({\rm Ric}+Kg)(\nabla v,\nabla v)+(\gamma-1)(\Delta v)^2\right]vu\,dV\notag\\
&+2(\gamma-1)\sigma_K\int_MK|\nabla v|^2vu\,dV+2\frac{\dot{\sigma}_K}{\sigma_K}\frac{d}{dt}\mathcal{N}_K
+\left(\frac{\ddot{\sigma}_K}{\sigma_K}
-\frac{2\dot{\sigma}_K^2}{\sigma_K^2}\right)\mathcal{N}_K\notag\\
\le&-2(\gamma-1)\sigma_K\int_M \left[|\nabla\nabla v|^2+({\rm Ric}+Kg)(\nabla v,\nabla v )+(\gamma-1)(\Delta v)^2\right]vu\,dV\notag\\
&+2\left(\frac{\dot{\sigma}_K}{\sigma_K}+{\kappa}\right)\frac{d}{dt}\mathcal{N}_K
+\left(\frac{\ddot{\sigma}_K}{\sigma_K}
-\frac{2\dot{\sigma}_K^2}{\sigma_K^2}-2{\kappa}\frac{\dot{\sigma}_K}{\sigma_K}\right)\mathcal{N}_K.\end{aligned}$$

Inspired by S. Li and X.-D. Li [@LiLi2](see also [@LiLi3] for a survey), we define the Perelman type $\mathcal{W}$-entropy by

$$\begin{aligned}
\label{KWentropy}
\mathcal{W}_K(t):=&\frac1{\dot{\alpha}_K(t)}\frac d{dt}(\alpha_K(t)\mathcal{N}_K(t))
=\mathcal{N}_K+\beta_K(t)\frac{d}{dt}\mathcal{N}_K\notag\\
=&-\sigma_K\int_M\Big[\beta_K(\gamma-1)\Delta v+\big(1+(\log\sigma_K)'\beta_K\big)\Big]vu\,dV,\notag\\
=&\sigma_K\beta_K\int_M\left[\gamma\frac{|\nabla v|^2}{v}-\left(\frac{1}{\beta_K}+\frac{\dot{\sigma}_K}{\sigma_K}\right)\right]vu\,dV,\end{aligned}$$

where $\beta_K(t)=\frac{\alpha_K}{\dot{\alpha}_K}$, then $$\frac{d}{dt}\mathcal{W}_K(t)=\beta_K\left(\frac{d^2}{dt^2}\mathcal{N}_K
+\frac{1+\dot{\beta}_K}{\beta_K}\frac{d}{dt}\mathcal{N}_K\right).$$ Combining and , we have

$$\begin{aligned}
\label{wkpment1}
\frac{d}{dt}\mathcal{W}_K(t)\le&
-2(\gamma-1)\sigma_K\beta_K\int_M \left[|\nabla\nabla v|^2+({\rm Ric}+Kg)(\nabla v,\nabla v )+(\gamma-1)(\Delta v)^2\right]vu\,dV\notag\\
&+2\beta_K\left(\frac{\dot{\sigma}_K}{\sigma_K}+\frac{1+\dot{\beta}_K}{2\beta_K}
+{\kappa}\right)\frac{d}{dt}\mathcal{N}_K
+\beta_K\left(\frac{\ddot{\sigma}_K}{\sigma_K}
-\frac{2\dot{\sigma}_K^2}{\sigma_K^2}
-2{\kappa}\frac{\dot{\sigma}_K}{\sigma_K}\right)\mathcal{N}_K.\end{aligned}$$

On th