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

Abstract

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