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CMFD, 2011, Volume 39, Pages 36–65 (Mi cmfd172)  

This article is cited in 16 scientific papers (total in 16 papers)

Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics

V. V. Vlasova, N. A. Rautianb, A. S. Shamaeva

a Moscow Lomonosov State University, Faculty of Mechanics and Mathematics, Moscow, Russia
b Russian Plekhanov Academy of Economics, Faculty of Economics and Mathematics, Moscow, Russia

Abstract: In the present paper, we study integrodifferential equations with unbounded operator coefficients in Hilbert spaces. The principal part of the equation is an abstract hyperbolic equation perturbed by summands with Volterra integral operators. These equations represent an abstract form of the Gurtin–Pipkin integrodifferential equation describing the process of heat conduction in media with memory and the process of sound conduction in viscoelastic media and arise in averaging problems in perforated media (the Darcy law).
The correct solvability of initial-boundary problems for the specified equations is established in weighted Sobolev spaces on a positive semiaxis.
Spectral problems for operator-functions are analyzed. Such functions are symbols of these equations. The spectrum of the abstract integrodifferential Gurtin–Pipkin equation is investigated.

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English version:
Journal of Mathematical Sciences, 2013, 190:1, 34–65

Bibliographic databases:

UDC: 517.929

Citation: V. V. Vlasov, N. A. Rautian, A. S. Shamaev, “Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics”, Partial differential equations, CMFD, 39, PFUR, M., 2011, 36–65; Journal of Mathematical Sciences, 190:1 (2013), 34–65

Citation in format AMSBIB
\by V.~V.~Vlasov, N.~A.~Rautian, A.~S.~Shamaev
\paper Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics
\inbook Partial differential equations
\serial CMFD
\yr 2011
\vol 39
\pages 36--65
\publ PFUR
\publaddr M.
\jour Journal of Mathematical Sciences
\yr 2013
\vol 190
\issue 1
\pages 34--65

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    This publication is cited in the following articles:
    1. N. A. Rautian, “On the Structure and Properties of Solutions of Integro-Differential Equations Arising in Thermal Physics and Acoustics”, Math. Notes, 90:3 (2011), 455–459  mathnet  crossref  crossref  mathscinet  isi
    2. V. V. Vlasov, N. A. Rautian, “Integrodifferential equations in viscoelasticity theory”, Russian Math. (Iz. VUZ), 56:6 (2012), 48–51  mathnet  crossref  mathscinet
    3. V. V. Vlasov, N. A. Rautian, A. S. Shamaev, “Analysis of operator models arising in problems of hereditary mechanics”, Journal of Mathematical Sciences, 201:5 (2014), 673–692  mathnet  crossref  mathscinet
    4. A. S. Shamaev, V. V. Shumilova, “O spektre odnomernykh kolebanii sloistogo kompozita s komponentami iz uprugogo i vyazkouprugogo materialov”, Sib. zhurn. industr. matem., 15:4 (2012), 124–134  mathnet
    5. Vlasov V.V. Rautian N.A., “Spectral Analysis and Representations of Solutions to Abstract Integrodifferential Equations”, Dokl. Math., 89:1 (2014), 34–37  crossref  mathscinet  zmath  isi  elib  scopus
    6. A. B. Muravnik, “Functional differential parabolic equations: integral transformations and qualitative properties of solutions of the Cauchy problem”, Journal of Mathematical Sciences, 216:3 (2016), 345–496  mathnet  crossref
    7. V. V. Vlasov, N. A. Rautian, “Properties of solutions of integro-differential equations arising in heat and mass transfer theory”, Trans. Moscow Math. Soc., 75 (2014), 185–204  mathnet  crossref  elib
    8. V. V. Vlasov, R. Perez Ortiz, “Spectral Analysis of Integro-Differential Equations in Viscoelasticity and Thermal Physics”, Math. Notes, 98:4 (2015), 689–693  mathnet  crossref  crossref  mathscinet  isi  elib
    9. D. A. Zakora, “Operator approach to the ilyushin model for a viscoelastic body of parabolic type”, Journal of Mathematical Sciences, 225:2 (2017), 345–381  mathnet  crossref
    10. V. V. Vlasov, N. A. Rautian, “Well-posedness and spectral analysis of integrodifferential equations arising in viscoelasticity theory”, Journal of Mathematical Sciences, 233:4 (2018), 555–577  mathnet  crossref
    11. V. V. Vlasov, N. A. Rautian, “Spektralnyi analiz integrodifferentsialnykh uravnenii v gilbertovom prostranstve”, Trudy seminara po differentsialnym i funktsionalno-differentsialnym uravneniyam v RUDN pod rukovodstvom A. L. Skubachevskogo, SMFN, 62, RUDN, M., 2016, 53–71  mathnet
    12. Vlasov V.V., Rautian N.A., “Study of Functional-Differential Equations With Unbounded Operator Coefficients”, Dokl. Math., 96:3 (2017), 620–624  crossref  zmath  isi  scopus
    13. A. V. Davydov, Yu. A. Tikhonov, “On Properties of the Spectrum of an Operator Pencil Arising in Viscoelasticity Theory”, Math. Notes, 103:5 (2018), 841–845  mathnet  crossref  crossref  mathscinet  isi  elib
    14. V. V. Vlasov, N. A. Rautian, “Issledovanie operatornykh modelei, voznikayuschikh v teorii vyazkouprugosti”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 64, no. 1, Rossiiskii universitet druzhby narodov, M., 2018, 60–73  mathnet  crossref
    15. V. V. Vlasov, N. A. Rautian, “A study of operator models arising in problems of hereditary mechanics”, J. Math. Sci. (N. Y.), 244:2 (2020), 170–182  mathnet  crossref  elib
    16. D. A. Zakora, “Predstavlenie reshenii odnogo integro-differentsialnogo uravneniya i prilozheniya”, Materialy Voronezhskoi zimneimatematicheskoi shkolySovremennye metody teorii funktsiii smezhnye problemy.28 yanvarya–2 fevralya 2019 g.Chast 2, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 171, VINITI RAN, M., 2019, 78–93  mathnet  crossref
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