 Exponential initial-velocity tail for the Klein-Gordon equation

For $u_{tt}=u_{xx}-u$ with $u(x,0)=0$ and $u_t(x,0)=e^{-|x|}$, the solution outside $|x|\le t$ is exactly $t{e}^{-|x|}$. On the positive half-line, $t{e}^{-x}$ satisfies both the equation and initial data; the finite propagation speed limits dependence at $x>t$ to positive initial points. Uniqueness on this domain of dependence proves the formula there, and reflection gives the negative half-line. The exterior tail is nonzero because the initial velocity is not compactly supported.