On a Boundary Value Problem for a Third Order Elliptic-hyperbolic Equation with Superposition Operators of the First and Second Orders in a Rectangular Domain
Abstract
This article proposes a method for solving a Dirichlet type problem for a third-order equation of elliptic-hyperbolic type with a superposition operators of first and second orders in a rectangular domain. It is shown that the correctness of the formulated problem significantly depends on the ratio of the sides of the rectangle from the hyperbolic part of the mixed domain. An example is given in which the well posed problem with homogeneous conditions has a nontrivial solution. A solution to the problem is constructed in the form of a sum of a series of eigenfunctions of the corresponding one-dimensional spectral problem. A criterion for the uniqueness of a solution is established. When justifying the uniform convergence of a series, the problem of small denominators arises. In this connection, estimates of small denominators about the distance from zero with the corresponding asymptotic have been established. These estimates made it possible to prove the convergence of the series in the class of regular solutions to this equation. Estimates on the stability of the solution from given boundary functions are proved.