Boundary-value problem with shift for a third-order parabolic-hyperbolic equation

In this paper, we study a boundary-value problem with shift for a third-order inhomogeneous parabolic-hyperbolic equation with the wave operator in the hyperbolicity domain. The boundary condition of the problem is the linear combination of the values of the unknown function on two independent characteristic lines and on the line of change of type. We obtain necessary and sufficient conditions for the existence and uniqueness of a regular solution of the problem. In some particular cases, explicit solutions are obtained.