This is a question coming to me when I was reading Qian Wang's Lectures on Nonlinear Wave Equation. Link below: http://people.maths.ox.ac.uk/wangq1/Lecture_notes/nonlinear_wave_9.pdf
Let $\square_g:=\partial_t^2-g^{ij}\partial_i\partial_j$, where $g$ is a $C^\infty$ symmetric matrix function on $[0,T]\times\mathbb{R}^n$ and is elliptic in the sense that there exist constants $0<\lambda\leq\Lambda<\infty$ s.t. for any $(t,x)\in[0,T]\times\mathbb{R}^n$, $$\lambda|\xi|^2\leq g^{ij}\xi_i\xi_j\leq\Lambda|\xi|^2,\xi\in\mathbb{R}^n.$$ (I guess equivalently here we can assume $g$ is a Riemannian metric for each time $t\in [0,T]$. Correct me if I am wrong. Thanks!)
A lemma in page 18 says the following,
For any $u\in C^2([0,T]\times\mathbb{R}^n)$, there holds $$\|\partial u(t,\bullet)\|_{L^2}\leq C_0(\|\partial u(0,\bullet)\|_{L^2}+\int_0^t\|\square_gu(\tau,\bullet)\|_{L^2}d\tau)\exp{(C_1\int_0^t\sum_{i,j}\|\partial g^{ij}(\tau,\bullet)\|_{L^2}d\tau)},$$ for all $t\in [0,T]$, where $C_0,C_1>0$ are dependent only on the ellipticity constants $\lambda,\Lambda$.
The notation $|\partial f|^2=|\partial_t f|^2+|\partial_x f|^2$, where $|\partial_x f|^2=|\partial_1 f|^2+\cdots+|\partial_n f|$, for $f=f(t,x)$
In the proof, an energy functional is defined as the following, $$E(t):=\int_{\mathbb{R}^n}(|\partial_t u|^2+g^{ij}\partial_i u\partial_j u)dx.$$
The whole proof makes sense to me except for the reason why $E(t)$ is well-defined, i.e., why the integral defining $E(t)$ converges.
I really appreciate your help.