The Dirichlet problem for a mixed-type equation with characteristic degeneration in a rectangular domain

Abstract

We study the first boundary-value problem for the following mixed-type equation of the second kind:

$$ u_{xx} + yu_{yy} + au_y - b^2 u = 0 $$

in the domain {(x, y) | 0 < x < 1, −α < y < β}, where a, b, α, and β are given real numbers, and 0 < a < 1, b ≥ 0, α > 0, β > 0. Based on the completeness of the system of eigenfunctions of one-dimensional spectral problem we establish a uniqueness criterion. We construct a solution to the problem as the sum of the series in eigenfunctions.