Criterion for a generalized solution in the class \(L_p\) for the wave equation to be in the class \(W_p^1\) (Q1945176)

This paper concerns generalized solutions to the wave equation \(\partial^2_t u= \partial^2_x u\) for \((x,t)\in Q_T:= [0,l]\times [0,T]\) with zero initial conditions and Dirichlet boundary conditions \(u(0,t)=\mu(t)\), \(u(1,t)= 0\) for \(t\in[0,T]\). The following result is proved: The solution belongs to the Sobolev space \(W^{1,p}(Q_T)\) if and only if the generalized derivative \(\mu'\) exists on \([0,T]\) and \[ \int^T_0 (T-t)|\mu'(t)|\,dt< \infty. \]