Classical solution of an initial-boundary value problem with a mixed boundary condition and conjugation conditions for a mildly quasilinear wave equation

We consider an initial-boundary value problem for a mildly quasilinear wave equation in the first quadrant in which we pose the Cauchy conditions on the spatial half-line, a mixed boundary condition on the time half-line, and conjugation conditions on some characteristics. We prove the uniqueness of the solution and establish the conditions under which a classical solution exists