Stability of a smooth flow in a Laval nozzle

A boundary-value problem for a non-linear second-order equation of mixed type in a cylindrical domain is considered. This problem simulates the development of small disturbances in a transonic flow of a chemical mixture in a Laval nozzle. The existence of a regular solution is proved with the help of a priori estimates for a corresponding linear problem and the contractive mapping theorem. The solution of the linear problem is constructed by the Galerkin method.