Let us consider the wave equation $$(\partial\_t^2 - \Delta)u = 0$$ on a domain $U$ with coercive homogeneous boundary conditions $$Bu|\_{\partial U} = 0$$ that make $-\Delta$ self-adjoint. My question is, how can we construct such boundary conditions that the wave equation does not have finite speed of propagation any more? Can it be done at all? Thanks in advance...



Finite speed of propagation of wave equation
The propagation speed is still finite because the following standard argument works independently of what happens at the boundary:


Assume that $u\in C^2$ solves the wave equation and $u(t=0,x)=0$ on $B\_r(0)$ for some $r>0$, such that $\overline{B}\_r\subset U$. Then $u(t,x)=0$ on $B\_{r-|t|}$. If we apply this to the difference $u=u\_2-u\_1$ of two solutions that agree on some open set, we obtain finite propagation speed.


To prove this claim (for $t\ge 0$), consider
$$
E(t) = \int\_{B\_{r-t}} (u\_t^2 + |\nabla u |^2)\, dx .
$$
Then
$$
E'(t) = -\int\_{S\_{r-t}} (u\_t^2 + |\nabla u |^2)\,d\sigma + 2\int\_{B\_{r-t}}
(u\_t u\_{tt} + \nabla u\_t \cdot \nabla u )\, dx .
$$
Integration by parts lets me rewrite
$$
\int\_{B\_{r-t}}\nabla u\_t \cdot \nabla u\, dx = -\int\_{B\_{r-t}} u\_t\Delta u\, dx
+ \int\_{S\_{r-t}} u\_t\, n\cdot \nabla u\, d\sigma ,
$$
where $n$ denotes the outer normal unit vector on the sphere. Thus, using the equation,
$$
E'(t) = \int\_{S\_{r-t}}(-u\_t^2-|\nabla u|^2+ 2 u\_t\, n\cdot \nabla u )\, d\sigma \le 0 ,
$$
so if $E(0)=0$, then we must have that $E\equiv 0$.


