# Unique continuation for the wave equation

Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $\Gamma \subset \subset \partial \Omega$, assume $u$ solves the wave equation

$$u_{tt}-c(x)\Delta u=0, \ \ u(x,0)=f(x),$$

where $f$ and $1-c$ are compactly supported in $\Omega$. Suppose $u(x,t)=0$ on $\Gamma \times (0,\infty)$. Can one guarantee that $u(x,t)=0$ in some open subset of $\mathbb{R}^n$ for all time (under some suitable assumptions on $\Gamma$)?

Try $u(x,t) = x_1 t$ with $\Gamma = \{x:\; x_1 = 0\}$.
• Thank you Robert. I just edited the question. I had an additional assumption that $u(x,0)$ has compact support which I had forgot to mention. – User4966 May 10 '18 at 20:14
• My $u(x,0) = 0$ and $1 - c(x) = 0$ have compact support. – Robert Israel May 11 '18 at 7:01
• I think the condition MathStudent wants is $u(\cdot,0)$ and $\partial_t u(\cdot,0)$ both have compact support. Then $u(\cdot,t)$ will have compact support at all times $t$. – Sam Zbarsky May 18 '18 at 20:57