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

Contemporary Mathematics. Fundamental Directions

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Contemporary Mathematics. Fundamental Directions, 2011, Volume 39, Pages 36–65 (Mi cmfd172)

This article is cited in 32 scientific papers (total in 32 papers)

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

V. V. Vlasov^a, N. A. Rautian^b, A. S. Shamaev^a

^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

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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.

English version:
Journal of Mathematical Sciences, 2013, Volume 190, Issue 1, Pages 34–65
DOI: https://doi.org/10.1007/s10958-013-1245-5

Bibliographic databases:

Document Type: Article

UDC: 517.929

Language: Russian

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

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\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.

\mathnet{http://mi.mathnet.ru/cmfd172}

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\transl

\jour Journal of Mathematical Sciences

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\crossref{https://doi.org/10.1007/s10958-013-1245-5}

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https://www.mathnet.ru/eng/cmfd/v39/p36

This publication is cited in the following 32 articles:

Andrey B. Muravnik, Grigorii L. Rossovskii, “Cauchy Problem with Summable Initial-Value Functions for Parabolic Equations with Translated Potentials”, Mathematics, 12:6 (2024), 895
I. V. Romanov, A. S. Shamaev, “On the existence of a wave front in the Cauchy problem for the Gurtin–Pipkin equation”, Math. Notes, 116:4 (2024), 862–866
O. V. Solonukha, “Nonlinear Differential-Difference Equations of Elliptic and Parabolic Type and Their Applications to Nonlocal Problems”, J Math Sci, 2024
O. V. Solonukha, “Nelineinye differentsialno-raznostnye uravneniya ellipticheskogo i parabolicheskogo tipa i ikh prilozheniya k nelokalnym zadacham”, SMFN, 69, no. 3, Rossiiskii universitet druzhby narodov, M., 2023, 445–563
Andrey B. Muravnik, “Differential-Difference Elliptic Equations with Nonlocal Potentials in Half-Spaces”, Mathematics, 11:12 (2023), 2698
Jian-Hua Chen, Lin Fu, Hua-Cheng Zhou, “Infinite-Time Admissibility of the Gurtin–Pipkin Systems in Hilbert Spaces”, SIAM J. Control Optim., 60:1 (2022), 505
I. V. Romanov, A. S. Shamaev, “Exact Control of a Distributed System Described by the Wave Equation with Integral Memory”, J Math Sci, 262:3 (2022), 358
V. V. Vlasov, N. A. Rautian, “Issledovanie integrodifferentsialnykh uravnenii metodami spektralnoi teorii”, Posvyaschaetsya pamyati professora N.D. Kopachevskogo, SMFN, 67, no. 2, Rossiiskii universitet druzhby narodov, M., 2021, 255–284
S. K. Zarifzoda, R. N. Odinaev, “Issledovanie nekotorykh klassov integro-differentsialnykh uravnenii v chastnykh proizvodnykh vtorogo poryadka so stepenno-logarifmicheskoi osobennostyu v yadre”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2020, no. 67, 40–54
V. V. Vlasov, N. A. Rautian, “Well-posedness and spectral analysis of integrodifferential equations of hereditary mechanics”, Comput. Math. Math. Phys., 60:8 (2020), 1322–1330
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
D. A. Zakora, “Predstavlenie reshenii odnogo integro-differentsialnogo uravneniya i prilozheniya”, Materialy Voronezhskoi zimnei matematicheskoi shkoly «Sovremennye metody teorii funktsii i smezhnye problemy». 28 yanvarya–2 fevralya 2019 g. Chast 2, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 171, VINITI RAN, M., 2019, 78–93
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
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
V. V. Vlasov, R. Perez Ortiz, N. A. Rautian, “Study of Volterra Integro-Differential Equations with Kernels Depending on a Parameter”, Diff Equat, 54:3 (2018), 363
V. V. Vlasov, N. A. Rautian, “Well-Posedness and Spectral Analysis of Volterra Integro-Differential Equations with Singular Kernels”, Dokl. Math., 98:2 (2018), 502
Vlasov V.V., Rautian N.A., “Study of Functional-Differential Equations With Unbounded Operator Coefficients”, Dokl. Math., 96:3 (2017), 620–624
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
N. A. Rautian, V. V. Vlasov, Springer Proceedings in Mathematics & Statistics, 164, Differential and Difference Equations with Applications, 2016, 411
V. V. Vlasov, N. A. Rautian, “Study of Volterra integro-differential equations arising in viscoelasticity theory”, Dokl. Math., 94:3 (2016), 639

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