Mild solution of a mixed problem for a third-order semilinear hyperbolic equation with the wave operator

Abstract

For a semilinear hyperbolic equation of the third-order given in the fjrst quadrant we study a mild solution of a mixed problem in which the initial conditions are specifjed on the spatial half-line and the mixed conditions are specifjed on the time half-line. The operator in the equation is a composition of the wave operator and the transport operator. Mild solution is defjned as a solution to coupled integral equations that are satisfjed 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 difgerentiable mild solution is the limit of classical solutions to the problem under study.