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On the Existence of Eigenvalues of a Boundary Value Problem with Transmitting Condition of the Integral Form for a Parabolic-Hyperbolic Equation

Author

Listed:
  • Abdumauvlen Berdyshev

    (Kazakh National Pedagogical University Named after Abai, Almaty 050010, Kazakhstan
    These authors contributed equally to this work.)

  • Alberto Cabada

    (Departamento de Estatística, Análise Matemática e Optimización, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia, Spain
    These authors contributed equally to this work.)

  • Erkinjon Karimov

    (Institute of Mathematics, Uzbekistan Academy of Sciences, Mirzo Ulugbek str., 81, Tashkent 100170, Uzbekistan
    These authors contributed equally to this work.)

Abstract

In the paper, we investigate a local boundary value problem with transmitting condition of the integral form for mixed parabolic-hyperbolic equation with non-characteristic line of type changing. Theorem on strong solvability of the considered problem has been proved and integral representation of the solution is obtained in a functional space. Using Lidskii Theorem on coincidences of matrix and spectral traces of nuclear operator and Gaal’s formula for evaluating traces of nuclear operator, which is represented as a product of two Hilbert-Schmidt operators, we prove the existence of eigenvalues of the considered problem.

Suggested Citation

  • Abdumauvlen Berdyshev & Alberto Cabada & Erkinjon Karimov, 2020. "On the Existence of Eigenvalues of a Boundary Value Problem with Transmitting Condition of the Integral Form for a Parabolic-Hyperbolic Equation," Mathematics, MDPI, vol. 8(6), pages 1-13, June.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:6:p:1030-:d:375265
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    References listed on IDEAS

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    1. Jorge Ferreira, 1996. "On weak solutions of semilinear hyperbolic-parabolic equations," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 19, pages 1-8, January.
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