Prove that the solution for the wave equation in $\Omega = \mathbb{R} \times [0, \infty)$
$$u(x,0)=\begin{cases} 1 & x \in (-\infty,-1) \\ 0 & x \in (-1,\infty) \end{cases} $$
is $u=0$ in the area $B=\{(x,t) | t \geq0,x-t \geq -1, x+t \leq 1\}$ but $u \neq 0$ outside of this area, i.e. $\{(x,t)|t \geq 0, t = 1 - x + \epsilon \lor t=x+1+ \epsilon\}, \epsilon > 0$.
For the first part I concluded that if $t \geq 0,x-t \geq -1, x+t \leq 1$ this means that $t=0,x=1$. So $u(x,0)=F(x)+G(x)=0$ (general solution of the wave equation). The $= 0$ is because $x \in (-1, \infty)$. This implies that $u(x,t)=F(x+t)+G(x-t)=F(x)+G(x)=0$ hence the first part is proven.
Is that correct? And how would I prove the second part?
