  Let $\pi:\mathbb{C}^n\times\mathbb{C}\rightarrow\mathbb{C}$ be the projection
map onto the second factor and let $D$ be a domain in $\mathbb{C}^{n+1}$ such
that for $y\in\pi(D)$, every fiber $D_y:=D\cap\pi^{-1}(y)$ is a smoothly
bounded strongly pseudoconvex domain in $\mathbb{C}^n$ and is diffeomorphic to
each other. By Chau's theorem, the K\"ahler-Ricci flow has a long time solution
$\omega_y(t)$ on each fiber $D_y$. This family of flows induces a smooth real
(1,1)-form $\omega(t)$ on the total space $D$ whose restriction to the fiber
$D_y$ satisfies $\omega(t)\vert_{D_y}=\omega_y(t)$. In this paper, we prove
that $\omega(t)$ is positive for all $t>0$ in $D$ if $\omega(0)$ is positive.
As a corollary, we also prove that the fiberwise K\"ahler-Einstein metric is
positive semi-definite on $D$ if $D$ is pseudoconvex in $\mathbb{C}^{n+1}$.
