Given $a_{ij}:\mathbb{R^n}\to \mathbb{R}$ and suppose that $a_0>0$ which $\sum_{ij}a_{ij}\xi_i\xi_j \geq a_o|\xi|^2$, $\forall x\in \mathbb{R^n}, \xi\in \mathbb{R^n}$. Considering $$\frac{\partial ^2}{\partial t^2}u(x,t)=\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{\partial}{\partial x_i}(a_{ij}(x)\frac{\partial u}{\partial x_j}(x,t))$$ I need to prove that, if $u\in C^2(\mathbb{R^n}\times [0,T)), T>0$ is a compact support function then $$E(t)=\int_{\mathbb{R^n}}\big[(\frac{\partial u}{\partial t})^2+\sum_{i=1}^{n}\sum_{j=1}^{n}(a_{ij}(x)\frac{\partial u}{\partial x_j}(x,t)\frac{\partial u}{\partial x_i}(x,t))\big]$$ is a constant for all $t\in[0,T)$
My attempt: we need to show that $E'(t)=0, \forall t\in[0,T)$. Then, I multiplicated the PDE by $\frac{1}{2}\frac{\partial u}{\partial t}$, then the first term can be manipulated and be $(\frac{\partial u}{\partial t})^2$. After this, the other term seems doesn't help in the manipulation (I think that I'm not seeing the manipulation). Some hint?