CLASSICAL SOLUTION OF A MIXED PROBLEM WITH DIRICHLET AND WENTZEL CONDITIONS FOR THE BIWAVE EQUATION WITH NONLINEAR LOWER ORDER 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 semi-axis, and Dirichlet and Wentzel conditions are specified on the temporal semi-axis. The solution is constructed by the method of characteristics in an implicit form as a solution of some integrodifferential 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.