Consider the PDE
\begin{align} u_{tt} - \nabla \cdot (c^2 \nabla u) + qu &= 0 \\ u(0,x) &= g(x) \\ u_t(0,x) &= h(x) \end{align}
where $c, q \geq 0$ depend only on $x$ and $0 < c_1 \leq c(x) < c_2$ for all $x \in \mathbb{R}^n$.
a) Fix $x_0 \in \mathbb{R}^n$ and $R_0 > 0$ and $t < R_0/c_2$. Let
\begin{equation} E(t)=\int_{B(x_0;R_0-c_2t)} \left(u_t^2 + c^2(x) \lvert \nabla u \rvert^2 + q(x) u^2 \right) d^nx \end{equation}
Show that $E(t)$ is decreasing.
b) Suppose that $\text{supp}(g), \text{supp}(h) \subset \{\lvert x \rvert < R \}$. Show that $u(t,x)=0 $ if $t > 0$ and $\lvert x \rvert > R + c_2 t$.
Showing a) seems easy as it is clear that $E(t_1) \geq E(t_2)$ for $0 \leq t_1 \leq t_2$ as increasing $t$ only decreases the radius of the ball over which we are integrating and the fact that the integrand is positive.
However, I am not sure how to deal with b). The approach that seems reasonable is to show that the energy is zero (which would imply $u$ is zero) for some initial moment and then use the fact that the energy is decreasing and non-negative to show that this holds for $t>0$ but still not sure.
In short, I'm looking for a hint in solving b).
