Solvability of Mixed Problems for the Klein–Gordon–Fock Equation in the Class Lp for p ≥ 1

We prove that the mixed problem for the Klein–Gordon–Fock equation utt(x, t) − uxx(x, t) + au(x, t) = 0, where a ≥ 0, in the rectangle QT = [0 ≤ x ≤ l] × [0 ≤ t ≤ T] with zero initial conditions and with the boundary conditions u(0, t) = μ(t) ∈ Lp[0, T ], u(l, t) = 0, has a unique generalized solution u(x, t) in the class Lp(QT) for p ≥ 1. We construct the solution in explicit analytic form. © 2018, Pleiades Publishing, Ltd.