Let $M$ be a complex compact manifold of dimension $n$ and consider the product $M\times [0,T)$. We are interested on the follwoing PDE:
$$ \frac{\partial u}{\partial t} = \log(\det (g_{j\overline k} + \partial^2_{j\overline k}u)) - \log \det g_{j\overline k} + f,$$ where $g_{j\overline k}$ and $f$ does not dependt on time.
We assume that $u(0) = 0$.
The paper claims that:
$$\max_M|\frac{\partial u}{\partial t}| \le \max_{M}|f|.$$
By priori estimates I could show that if $v := u -\frac{1}{vol(M)}\int_M udV$ then there exists a constant $C > 0$ such that $$\|v\|_{C^{\infty}(M\times [0,T)} \le C.$$
The paper claims that there is a sequence $t_n \to \infty$ such that $v(x,t_n) \to v(x,\infty)$ on the $C^{\infty}$ topology, and since $\max |\frac{\partial u}{\partial t}u|$ is bounded for every time, then the solution cannot blow up int finite time.
How can I show that there exists such sequence? How limitation of $\frac{\partial u}{\partial t}$ implies that solution does not blow up in finite time?