Consider the family of equations (1.7), and assume that $f(\lambda)$ satisfies the structural condition (1.4). Let $\varphi_{t}$ be $C^{2}$ functions on $X$ satisfying (1.7). Fix $p>n$. Then for any $t\in(0,1]$, we have

$\displaystyle\sup_{X}|\varphi_{t}|\leq C$

(1.10)

where $C$ is a constant depending only on $\omega_{X},\chi,p,n,\gamma$, and upper bounds for the following three quantities

$\displaystyle{c_{t}^{n}\over V_{t}},\quad E_{t}(\varphi_{t}),\quad{\rm Ent}_{p}(F_{t}).$

(1.11)