led with Ricci flow $$\label{adjoint}
\partial_tg=-2{\rm Ric},\quad\partial_t f=-\Delta f+|\nabla f|^2+R+\frac n{2\tau},\quad\partial_t\tau=-1.$$ Perelman [@P] introduced the following $\mathcal{W}$-entropy $$\mathcal{W}(g,f,\tau):=\int_M\Big(\tau(R+|\nabla f|^2)+f-n\Big)\frac{e^{-f}}{(4\pi\tau)^{\frac n2}}\,dV$$ and proved its monotonicity $$\label{Pentropy}
\frac{d}{dt}\mathcal{W}(g,f,\tau)=2\tau\int_M\Big|R_{ij}+\nabla_i\nabla_jf-\frac1{2\tau}g_{ij}\Big|^2\frac{e^{-f}}{(4\pi\tau)^{\frac n2}}\,dV\ge0,$$ where $\tau>0$ and $\int_M{(4\pi\tau)^{-\frac n2}}e^{-f}dV=1$. The static point of $\mathcal{W}$-entropy is called gradient shrinking Ricci soliton $$\label{soliton}
R_{ij}+\nabla_i\nabla_jf=\frac1{2\tau}g_{ij},$$ which can be viewed as a model with singularity when studying the singularity formation of solutions of the Ricci flow. Moreover, when $H(x,t)={(4\pi\tau)^{-\frac n2}}e^{-f}$ is a fundamental solution to the conjugate heat equation , Perelman’s differential Harnack inequality holds [@P; @NiLYH] $$\label{PLYH}
v_H:=\Big(\tau(R+2\Delta f-|\nabla f|^2)+f-n\Big)H\le0.$$

A feature of Perelman’s entropy monotonicity formula is that it holds at any dimension and without assumption of curvature condition. After Perelman’s work, an interesting direction is that of finding entropy monotonicity formulae for other geometric evolution equations. There were some development in this direction, L. Ni [@Nientropy] derived the entropy monotonicity formula for the linear heat equation on Riemannian manifolds with nonnegative Ricci curvature, $$\label{Nientropy}
\frac{d}{dt}\mathcal{W}(f,\tau)=-2\tau\int_M\left(\Big|\nabla_i\nabla_jf-\frac1{2\tau}g_{ij}\Big|^2+R_{ij}f_if_j\right)u\,dV,$$ where $u=(4\pi\tau)^{-\frac n2}e^{-f}$ is a positive solution to the heat equation $\partial_tu=\Delta u$ with $\int_Mu\,dV=1$, $\frac{d\tau}{dt}=1$ and $\mathcal{W}(f,\tau)$ is defined by $$\mathcal{W}(f,\tau):=\int_M\Big(\tau|\nabla f|^2+f-n\Big)u\,dV.$$

When the Ricci curvature of $M$ is bounded below, Li-Xu [@LX] established s