Initial-Boundary Value Problem for a Third-Order Semilinear Hyperbolic Equation with the Wave Operator

For a semilinear hyperbolic equation of the third order given in the first quadrant, we study an initial-boundary value problem in which the initial conditions are specified on the spatial half-line and the mixed 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 some integral equations. We study the solvability of these equations depending on the initial data and their smoothness. For the problem under consideration, the uniqueness of the solution is proved, and conditions are established under which there exists a classical solution.