Classical solution of a mixed problem with the Dirichlet and Neumann conditions for a nonlinear biwave equation

For a nonlinear biwave equation given in the first quadrant, we consider a mixed problem in which the Cauchy conditions are specified on the spatial half-line, and the Dirichlet and Neumann conditions are specified on the time half-line. The solution is constructed by the method of characteristics in an implicit analytical form as a solution of a certain integro-differential equations. By the method of continuation with respect to a parameter and a priori estimates, the solvability of these equations, the dependence on the initial data, and the smoothness of solutions are examined. For the problem considered, the uniqueness of the solution is proved and the conditions of the existence of classical solution are established. If the matching conditions are not met, then a problem with conjugation conditions is constructed, and if the data is not sufficiently smooth, then a mild solution is constructed.