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Journal of Mathematical Sciences

, Volume 236, Issue 6, pp 603–640 | Cite as

Initial-Boundary-Value Problem for Inhomogeneous Degenerate Equations of Mixed Parabolic-Hyperbolic Type

  • K. B. Sabitov
  • S. N. Sidorov
Article
  • 5 Downloads

Abstract

We consider initial-boundary-value problems for three classes of inhomogeneous degenerate equations of mixed parabolic-hyperbolic type: mixed-type equations with degenerate hyperbolic part, mixed-type equations with degenerate parabolic part, and mixed-type equations with power degeneration. In each case, we state a criterion of uniqueness of a solution to the problem. We construct solutions as series with respect to the system of eigenfunctions of the corresponding one-dimensional spectral problem. We prove that the uniqueness of the solution and the convergence of the series depend on the ratio of sides of the rectangular from the hyperbolic part of the mixed domain. In the proof of the existence of solutions to the problem, small denominators appear that impair the convergence of series constructed. In this connection, we obtain estimates of small denominators separated from zero and the corresponding asymptotics, which allows us, under certain conditions, to prove that the solution constructed belongs to the class of regular solutions.

Keywords and phrases

mixed parabolic-hyperbolic equation initial-boundary-value problem spectral method uniqueness existence series small denominators uniform convergence 

AMS Subject Classification

35M10 35Q60 

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Applied Studies of the Republic BashkortostanSterlitamakRussia

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