The kaehlerian Einstein flow

Question

Let $(M^n,\omega)$ be a Kaehler manifold with kaelerian form $\omega$ and Ricci form $\rho$, and let $\lambda$ be a function. The kaehlerian Einstein equations are:

$$\rho = \lambda \omega +n \partial \bar{\partial} (log (\lambda))$$

Have we solutions for the kaehlerian Einstein equations?

We can also define a kaehlerian Einstein flow:

$$\frac{\partial \omega}{\partial t}=\rho-\lambda \omega - n\partial \bar{\partial}(log(\lambda ))$$

$\lambda$ may be the scalar curvature. Have we solutions for this flow at short time? Does the flow converge towards a Kaehler-Einstein metric?