In a paper by Zuazua, "Propagation, Observation, Control, and Numerical Approximation of Waves", which you can find here http://www.sissa.it/fa/am/DCS2003/reading_mat/zuazuajpg.pdf he considers the wave equation
\begin{equation} \begin{cases} u_{tt} - u_{xx} =0,\\ u(0,t) = 0, u(1,t) = v(t),\\ u(x,0) = u_0, u_t(x,0) = u_1, \end{cases} \end{equation}
where $u_0 \in L^2(0,1), u_1 \in H^{-1}(0,1), v\in L^2(0,1)$.
Why does this have a solution with these initial conditions? Furthermore, why is $u(1,t)$ even defined? It seems this is supposed to be defined pointwise... but here this is only $H^{-1}$ !