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 $u(0) = 0$.
Then the paper I am reading claim (without prove, just saying that this is a Maximum Princinple for Parabolic equations) that
$$\max_M|\frac{\partial u}{\partial t}| \le \max_{M}|f|.$$
How can I prove this claim? Every maximum principle I know wasnt the right to prove this.