Classical Solution to a Mixed Problem with the Dirichlet andWentzel Conditions for the Biwave Equationwith Nonlinear Lower Terms

For the hyperbolic biwave equation with nonlinear lower terms given in the first quadrant of Euclidean space, we consider a mixed problem in which the Cauchy conditions are specified on the spatial half-line, and the Dirichlet and Wentzel conditions are specified on the temporal half-line. The solution is constructed by the method of characteristics in implicit form as a solution to some integro-differential equations. The solvability of these equations, as well as the dependence on the initial data and the smoothness of their solutions, is studied using the parameter continuation method and a priori estimates. For the problem under consideration, the uniqueness of the solution is proved and conditions under which there exists a 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 smooth enough, then a mild solution is constructed.