  Let $(M,g(t))$, $0\le t\le T$, $\partial M\ne\phi$, be a compact
$n$-dimensional manifold, $n\ge 2$, with metric $g(t)$ evolving by the Ricci
flow such that the second fundamental form of $\partial M$ with respect to the
unit outward normal of $\partial M$ is uniformly bounded below on $\partial
M\times [0,T]$. We will prove a global Li-Yau gradient estimate for the
solution of the generalized conjugate heat equation on $M\times [0,T]$. We will
give another proof of Perelman's Li-Yau-Hamilton type inequality for the
fundamental solution of the conjugate heat equation on closed manifolds without
using the properties of the reduced distance. We will also prove various
gradient estimates for the Dirichlet fundamental solution of the conjugate heat
equation.
