Flow Convergence from Perturbation of Elliptic Solution

Suppose $(M,\omega)$ is a complete Kaher manifold with bounded geometry, and $u\in C^{\infty}(M)$ solves

$$(\omega+\sqrt{-1}\partial\overline{\partial}u)^{n}=e^{f}\omega^{n},$$

and satisfies

$$C^{-1}\omega\leq\omega+\sqrt{-1}\partial\overline{\partial}u\leq C\omega$$ $$|f|(z)\leq\frac{C}{d_{\omega}^{s}(z_{0},z)}$$

as $d_{\omega}(z_{0},z)\to\infty$, for some fixed point $z_{0}\in M$. Then does the flow

$$\partial_{t}u=\log\left(\frac{(\omega+\sqrt{-1}\partial\overline{\partial}u)^{n}}{\omega^{n}}\right)-f$$

have long time existence? Does it converge to a solution of the corresponding elliptic equation?

This is known when $s>2$. It is also known by Li-Tam that a longtime solution exists assuming $(M,\omega_{0})$ is a complete Kahler manifold with nonnegative holomorphic bisectional curvature and maximal volume growth. Moreover, in this case, we have

$$|Rm|_{\omega_{t}}\leq\frac{C}{t}$$

for all $t>0$. However, it may not be known whether $\omega_{t}\to\omega$ as $t\to\infty$.

There is always a solution of $(\omega+\sqrt{-1}\partial\overline{\partial}u)^{n}=e^{f}\omega^{n}$ if $s\in(0,2)$ and $(M,\omega)$ is asymptotically conical.