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

Abdumauvlen Berdyshev, Alberto Cabada and Erkinjon Karimov
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Abdumauvlen Berdyshev: Kazakh National Pedagogical University Named after Abai, Almaty 050010, Kazakhstan
Alberto Cabada: Departamento de Estatística, Análise Matemática e Optimización, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia, Spain
Erkinjon Karimov: Institute of Mathematics, Uzbekistan Academy of Sciences, Mirzo Ulugbek str., 81, Tashkent 100170, Uzbekistan

Mathematics, 2020, vol. 8, issue 6, 1-13

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.

Keywords: transmitting condition; parabolic-hyperbolic equation; Green’s function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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