  We study a parabolic complex Monge-Amp\`{e}re type equation of the form
\eqref{MA} on a complete noncompact \K manifold. We prove a short time
existence result and obtain basic estimates. Applying these results, we prove
that under certain assumptions on a given real and closed (1,1) form $\Omega$
and initial \K metric $g_0$ on $M$, the modified \KR flow $g'=-\Ric+\Omega$ has
a long time smooth solution converging to a complete \K metric such that
$\Ric=\Omega$, which extends the result in [1] to non-compact manifolds. We
will also obtain a long time existence result for the \KR flow which
generalizes a result [5].
