Mixed problem for the wave equation with integrable potential in the case of two-point boundary conditions of distinct orders

Abstract

We study a mixed problem for the wave equation with integrable potential and with two-point boundary conditions of distinct orders for the case in which the corresponding spectral problem may have multiple spectrum. Based on the resolvent approach in the Fourier method and the Krylov convergence acceleration trick for Fourier series, we obtain a classical solution u(x, t) of this problem under minimal constraints on the initial condition u(x, 0) = ϕ(x). We use the Carleson–Hunt theorem to prove the convergence almost everywhere of the formal solution series in the limit case of ϕ(x) ∈ L p[0, 1], p > 1, and show that the formal solution is a generalized solution of the problem.