Initial-boundary value problem with Dirichlet and Wentzell conditions for a mildly quasilinear biwave equation

For a nonstrictly hyperbolic mildly quasilinear biwave equation in the first quadrant, an initial-boundary value problem with the Cauchy conditions specified on the spatial half-line and the Dirichlet and Wentzell conditions applied on the time half-line was examined. The solution was constructed in an implicit analytical form as a solution of some integro-differential equations. The solvability of these equations was investigated using the parameter continuation method. For the problem under study, the uniqueness of the solution was proved, and the conditions under which its classical solution exists were established. In the case when the data were not smooth enough, a mild solution was constructed.