On a modified parabolic complex Monge-Amp\`{e}re equation with applications

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 $\mathrm{\Omega }$ and initial \K metric $g_0$ on $M$, the modified \KR flow $g^{\mathrm{\prime }}=-\text{Ric}+\mathrm{\Omega }$ has a long time smooth solution converging to a complete \K metric such that $\text{Ric}=\mathrm{\Omega }$, 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].