On the Dirichlet Problem for a Three-dimensional Equation of Mixed Type with Two Planes of Type Change in a Parallelepiped

The first boundary value problem for the three-dimensional model equation with two planes of type change in a parallelepiped is investigated. A criterion for the uniqueness of its solution is established. The solution of problem is constructed as the sum of the series in the biorthogonal system of the corresponding spectral problem for an ordinary differential operator. When proving the existence of a solution, a problem of small denominators arises. Therefore, are obtained estimates on the separation of these denominators from zero with the corresponding asymptotics. This allows us to establish the existence of a solution to the problem from the required class.