  In this article we study the K\"ahler Ricci flow, the corresponding parabolic
Monge Amp\`{e}re equation and complete non-compact K\"ahler Ricci flat
manifolds. In our main result Theorem \ref{mainthm} we prove that if $(M, g)$
is sufficiently close to being K\"ahler Ricci flat in a suitable sense, then
the K\"ahler Ricci flow \eqref{KRF} has a long time smooth solution $g(t)$
converging smoothly uniformly on compact sets to a complete K\"ahler Ricci flat
metric on $M$. The main step is to obtain a uniform $C^0$-estimates for the
corresponding parabolic Monge Amp\`{e}re equation. Our results on this can be
viewed as a parabolic version of the main results in \cite{TY3} on the elliptic
Monge Amp\`{e}re equation.
