<p>We consider an initial-boundary-value problem for a semilinear wave equation in the first quadrant in which we pose the Cauchy conditions on the spatial half-line and the Zaremba boundary condition on the time half-line. We reformulate this problem as a problem with conjugation conditions on the characteristics. The imposed inhomogeneous conjugation conditions uniquely define discontinuity of the solution on the characteristics. By the method of characteristics, we construct a solution in an implicit analytical form as a solution of some integro-differential equations. The solvability of these equations and the dependence on the initial data and the smoothness of their solutions are examined. For the problem considered, we prove the uniqueness of a solution and establish conditions of the existence of a classical solution. A mild solution is constructed in the case of insufficiently smooth data of the problem. The obtained mathematical results are applied to a problem from combustion theory.</p>