The existence of solution of wave equation with initial/boundary data in $H^{-1}$ and $L^2$

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

$$\begin{cases} u_{tt} - u_{xx} =0,\\ u(0,t) = 0, u(1,t) = v(t),\\ u(x,0) = u_0, u_t(x,0) = u_1, \end{cases}$$

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 $H^{-1}$ !

• @NormalHuman Ok what would you suggest? – MarsOneRover Nov 8 '15 at 22:17
• These conditions cannot be interpreted in the classical (pointwise) sense. They can be enforced in the weak sense, as boundary conditions under integration by parts; as is done in the paper. – user147263 Nov 8 '15 at 22:38
• @NormalHuman It does not seem to be done in the paper, and he makes no remarks as to it... at least from what I can see. – MarsOneRover Nov 8 '15 at 22:49
• This solution has been obtained by the transposition sense (passing by the adjoint problem), it is a simple consequence of the Riesz representation theorem. For instance, see the Lions book. – Gustave Apr 3 '18 at 13:41