Initial-Boundary Problem for a Three-Dimensional Inhomogeneous Equation of Parabolic-Hyperbolic Type

Abstract For an inhomogeneous three-dimensional equation of mixed parabolic-hyperbolic type in a rectangular parallelepiped, the initial-boundary problem is studied. A criterion for the uniqueness of a solution is established. The solution is constructed as the sum of an orthogonal series. In substantiating the convergence of the series, the problem of small denominators of two natural arguments arose. Estimates are established for the separation from zero of the small denominators with the corresponding asymptotics. These estimates made it possible to justify the convergence of the constructed series in the class of regular solutions of this equation.