Mixed problem for a homogeneous wave equation with a nonzero initial velocity and a summable potential

For a mixed problem defined by a wave equation with a summable potential equal-order boundary conditions with a derivative and a zero initial position, the properties of the formal solution by the Fourier method are investigated depending on the smoothness of the initial velocity $u_t '(x, 0)=\psi (x)$. The research is based on the idea of A. N. Krylov on accelerating the convergence of Fourier series and on the method of contour integrating the resolvent of the operator of the corresponding spectral problem. The classical solution is obtained for $\psi (x)\in W_p^1$ ($1 <p\le 2$), and it is also shown that if $\psi(x)\in L_p[0,1]$ ($1\le p\le2$), the formal solution is a generalized solution of the mixed problem.