Mild solution of a mixed problem for a third-order semilinear hyperbolic equation with the wave operator - Korzyuk, Viktor I.; Rudzko, Jan V. (Journal of the Belarusian State University. Mathematics and Informatics, 2026)
For a semilinear hyperbolic equation of the third-order given in the first quadrant we study a mild solution of a mixed 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 operator in the equation is a composition of the wave operator and the transport operator. Mild solution is defined as a solution to coupled integral equations that are satisfied by a classical solution. It is shown that under some smoothness conditions on the initial and boundary data the problem under consideration admits the existence and uniqueness of the mild solution. It is established that the twice continuously differentiable mild solution is the limit of classical solutions to the problem under study.