Keldysh problem for a three-dimensional equation of mixed type with three singular coefficients in a semi-infinite parallelepiped

Abstract

This article studies the Keldysh problem for a three-dimensional equation of mixed type with three singular coefficients in a semi-infinite parallelepiped. Based on the completeness property of eigenfunction systems of two one-dimensional spectral problems, the uniqueness theorem is proved. To prove the existence of a solution to the problem, the Fourier spectral method based on the separation of variables is used. The solution to this problem is constructed in the form of a sum of a double Fourier-Bessel series. In substantiating the uniform convergence of the constructed series, we used asymptotic estimates of the Bessel functions of the real and imaginary argument. Based on them, estimates were obtained for each member of the series, which made it possible to prove the convergence of the series and its derivatives to the second order inclusive, as well as the existence theorem in the class of regular solutions.