Integral identity definition of a generalized solution in the class \(L_p\) to a mixed problem for the wave equation (Q1945216)

The authors show the equivalence of two definitions of generalized solutions to the initial-boundary value problem \(\partial^2_t u= \partial^2_x u\) for \((x,t)\in [0,l]\times [0,T]\), \(u= \partial_t u= 0\) for \(t=0\), \(u(0,t)= \mu(t)\), \(u(l,t)= 0\) for \(t\in [0,T]\) under the assumption that \[ \int^T_0 (T-t)|\mu(x)|^p\,dt< \infty. \]