Dirichlet problem for three-dimensional hyperbolic-parabolic equations with type and order extinction

Abstract

Boundary value problems for hyperbolic-parabolic equations in the plane have been studied
properly, where Tricomi problem and the first boundary value problem were investigated.
The mixed problem, Cauchy characteristic problem and Darboux problem for multidimensional
hyperbolic-parabolic equations have been considered before. Different authors have defined and
investigated Tricomi problem for hyperbolic-parabolic equations in multidimensional domains. The
theory of boundary value problems for degenerating hyperbolic-parabolic equations in the plane has
also been studied properly. Besides, multidimensional analogues of these problems in generalized
spaces have been investigated. Correctness of Dirichlet problems for degenerating multidimensional
hyperbolic equations has been proved. In this work the author showed solvability and obtained
an explicit classical solution of Dirichlet problem in a cylindrical domain for three-dimensional
hyperbolic-parabolic equations with type and order extinction.