Suppose that $u$ is a $C^2$ solution to the wave equation in $\mathbb{R}^n\times\mathbb{R}$
Show that if $u(\cdot,0)$ and $u_t(\cdot,0)$ have compact support in $\mathbb{R}^n$, then $u(\cdot,t)$ has compact support in $\mathbb{R}^n$ for all $t≥0$ . Furthermore, how is this false if only $u(\cdot, 0)$ has compact support.
My hint from class was to prove that $u(\cdot,t)$ is supported in $B(0,R+t)$ for any $t>0$. If you pick a point $x$ that is not in $B(0,R+t)$, then your goal is to prove that $u(x,t)=0$. This can be accomplished using the Finite Propagation Speed Theorem that we proved in class.