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

Abstract

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 and a mixed boundary condition on the time half-line. We use the method of characteristics to construct the solution in an implicit analytical form as a solution of some integro-differential equations. The solvability of these equations is studied, as well as the dependence on the initial data and the smoothness of their solutions. For the problem in question, the uniqueness of the solution is proved, and the conditions under which its classical solution exists are established