Assume $a^{ij}$, $i, j = 1, 2$ are smooth functions on $\mathbb{R}^2$ such that $a^{ij} = a^{ji}$. Write $A = (a^{ij})$ for the $2$x$2$ matrix with $a^{ij}$ as entries, and assume there exists constants $c, C > 0$ such that:
$c|\zeta|^2 \le \sum_{i, j = 1}^2 \langle A(x)\zeta, \zeta \rangle \le C|\zeta|^2$, for all $(x, \zeta) \in \mathbb{R}^2$ x $\mathbb{R}^2$.
Let $u$ be a smooth solution to the variable-coefficient wave equation $\partial_t ^2 u$ - div$(A \nabla u) = 0$,
such that $u$ and its derivatives decay rapidly as $|x| \rightarrow \infty$ (for instance, bounded by $|x|^{-3/2 - \epsilon}$ for some $\epsilon > 0$).
Show that, if $u(0, x) = 0$ and $\partial_t u(0, x) = 0$ for all $|x| \le 1$, then $u(t, x) = 0$ in the region
{$(t, x): t \ge 0, |x| \le 1 - \sqrt{C} t$} $\subset \mathbb{R}_t$ x $\mathbb{R}_x ^2$.
I think I kind of have an idea on how to start this problem, but I'm not sure if it's the right approach. Do I have to integrate both sides with respect to $t$ and $x$, and then use integration by parts to try to simplify things? I honestly don't really know where to start. Any help would be greatly appreciated. Thank you.