Uniqueness of solutions to the wave equation

Question

we are given the problem $u_{tt}-c^2\Delta u=0$ with conditions $u(x,0)=u_0(x)$ and $u_t(x,0)=u_1(x)$ where $x\in\mathbb{R}^n$ and $u_0,u_1\in\mathcal{C}^1$ having compact supports. If a solution exists, is it possible to conclude that this solution has a compact support, too? Thanks.

Answer 1

Both uniqueness and finite propagation speed follow by an energy method just like in the one-dimensional case. For the sake of simplicity the following is for $c=1$, the same argument works for arbitrary $c>0$.

Fix some point $x_0 \in \mathbb{R}^n$ and time $t_0 > 0$, and define $$ E(t) = \int_{B(x_0, t_0-t)} \left( u_t(x,t)^2 + |\nabla u (x,t)|^2 \right) \, dx $$ for $0 \le t \le t_0$. Using integration by parts (Green's identities) you should get $$ E'(t) = \int_{\partial B(x_0, t_0-t)} \left( 2 u_t \frac{\partial u}{\partial n} -u_t^2 - |\nabla u|^2\right) \, dx $$ Then, using the inequality $$ \left| 2u_t \frac{\partial u}{\partial n} \right| \le 2|u_t| \, |\nabla u| \le u_t^2 + |\nabla u|^2 $$ you get that $E'(t) \le 0$. So if you know $u(x,0)\equiv u_t(x,0)\equiv 0$ for $x \in B(x_0, t_0)$, then $E(0)=0$, and so $E(t)=0$ for $0 \le t \le t_0$, implying that $u_t(x,t) = 0$ and $\nabla u (x,t) = 0$ for $0\le t \le t_0$ and $|x-x_0| \le t_0 - t$, which implies $u(x,t) = 0$ for the same parameters. This directly implies uniqueness and propagation speed $\le 1$.