Let $\phi\in C^\infty([0,T],\mathbb{R}^n)$ be a solution of linear wave equation $$\sum_{\alpha,\beta=0}^ng^{\alpha\beta}\partial_\alpha\partial_\beta\phi=F$$ where $\partial_\alpha:=\frac{\partial}{\partial x_\alpha}, \alpha=0,1,2,\cdots,n$, $t=x_0$, $g^{\alpha\beta}, F$ are smooth functions in $[0,T]\times\mathbb{R}^{n}$, $g^{\alpha\beta}$ is symmetric and $$\sup_{[0,T]\times\mathbb{R}^n}\sum_{\alpha,\beta}|g^{\alpha\beta}-\eta^{\alpha\beta}|<\frac{1}{m}\ \ (m\in \mathbb R)$$ where $\eta_{00}=-1, \eta_{ii}=1, i=1,2,\cdots,n$ and $\eta_{\alpha i}=0$ otherwise, $|\partial f|^2:=\sum_{\alpha=0}^n|\partial_\alpha f|^2$.
How to show that there is a constant $C$ such that for any $t\in[0,T]$, $$\|\partial\phi(t,\cdot)\|_{L^2(\mathbb{R}^n)}\le C\left(\|\partial\phi(0,\cdot)\|_{L^2(\mathbb{R}^n)}+\int_0^t\|F(s,\cdot)\|_{L^2(\mathbb{R}^n)}\mathrm{d}s\right)$$ $$\times\exp\left(\int_0^t\sum_{\alpha,\beta}\|\partial g^{\alpha\beta}(s,\cdot)\|_{L^2(\mathbb{R}^n)}\mathrm{d}s\right)?$$
