Correctness of the mixed problem for degenerate multidimensional elliptic-parabolic equations

The initial-boundary value problem (Dirichlet problem) for general elliptic-parabolic equations of second order was first posed by G. Fichera. Further investigation of this problem was carried out in the monograph by O.A. Oleinik and E.V. Radkevich and the works by V.N. Vragov. In these works, the authors examined mixed problems for degenerate multidimensional elliptic equations. The articles by S.A. Aldashev focused on the correctness (in the sense of uniqueness of solvability) of the Dirichlet problem in a cylindrical domain for multidimensional elliptic-parabolic equations. A mixed problem for these equations has not been studied. In this paper, the authors demonstrate the uniqueness of solvability and obtain an explicit representation of the classical solution of the mixed problem for degenerate multidimensional elliptic-parabolic equations. The proposed method allows reducing the problem under study to a mixed problem for a degenerate multidimensional elliptic equation examined by S.A. Aldashev.