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Journal of Mathematical Sciences

, Volume 230, Issue 5, pp 668–672 | Cite as

Spectral Analysis of Linear Models of Viscoelasticity

  • V. V. Vlasov
  • N. A. Rautian
Article

Abstract

In this paper, we examine Volterra integrodifferential equations with unbounded operator coefficients in Hilbert spaces. Equations considered are abstract hyperbolic equations perturbed by terms containing Volterra integral operators. These equations can be realized as partial integrodifferential equations that appear in the theory of viscoelasticity (see [2, 5]), as Gurtin–Pipkin integrodifferential equations (see [1, 7]) that describe finite-speed heat transfer in materials with memory. They also appear in averaging problems for multiphase media (Darcy’s law.

Keywords and phrases

integrodifferential equation spectral analysis operator-valued function 

AMS Subject Classification

34D05 34C23 

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Notes

References

  1. 1.
    M. E. Gurtin and A. C. Pipkin, “Theory of heat conduction with finite wave speed,” Arch. Rat. Mech. Anal., 31, 113–126 (1968).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. A. Il’yushin and B. E. Pobedrya, Foundations of Mathematical Theory of Thermoviscoelasticity [in Russian], Nauka, Moscow (1970).Google Scholar
  3. 3.
    S. Ivanov and L. Pandolfi, “Heat equations with memory: lack of controllability to rest,” J. Math. Anal. Appl., 355, 1–11 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin (1984).Google Scholar
  5. 5.
    N. D. Kopachevsky and S. G. Krein Operator Approach to Linear Problems of Hydrodynamics. Vol. 2. Nonself-adjoint Problems for Viscous Fluids, Birkhäuser, Boston (2003).Google Scholar
  6. 6.
    J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin–Heidelberg–New York (1972).CrossRefzbMATHGoogle Scholar
  7. 7.
    A. V. Lykov, Problems of Heat and Mass Transfer [in Russian], Nauka i Tekhnika, Minsk (1976).Google Scholar
  8. 8.
    A. I. Miloslavsky, Spectral properties of an operator pencil that appears in the theory of viscoelasticity [in Russian], preprint, Kharkov (1987).Google Scholar
  9. 9.
    L. Pandolfi, “The controllability of the Gurtin–Pipkin equations: A cosine operator approach,” Appl. Math. Optim., 52, 143–165 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    A. A. Shkalikov, “Strongly damped pencils of operators and solvability of the corresponding operator-differential equations,” Mat. Sb., 135 (177), No. 1, 96–118 (1988).Google Scholar
  11. 11.
    V. V. Vlasov, A. A. Gavrikov, S. A. Ivanov, D. Yu. Knyazkov, V. A. Samarin, and A. S. Shamaev, “Spectral properties of combined media,” Sovr. Probl. Mat. Mekh., 5, No. 1, 134–155 (2009).Google Scholar
  12. 12.
    V. V. Vlasov, D. A. Medvedev, and N. A. Rautian, “Functional-differential equations in Sobolev spaces and their spectral analysis,” Sovr. Probl. Mat. Mekh., 8 (2011).Google Scholar
  13. 13.
    V. V. Vlasov and N. A. Rautian, “Correct solvability and spectral analysis of integrodifferential equations that appear in the theory of viscoelasticity,” Sovr. Mat. Fundam. Napr., 58, 22–42 (2015).Google Scholar
  14. 14.
    V. V. Vlasov and N. A. Rautian, “Correct solvability and spectral analysis of abstract hyperbolic integrodifferential equations,” Tr. Semin. Petrovskogo, 28, 75–114 (2011).zbMATHGoogle Scholar
  15. 15.
    V. V. Vlasov, N. A. Rautian, and A. S. Shamaev, “Spectral analysis and correct solvability of abstract integrodifferential equations that appear in thermal physics and acoustics,” Sovr. Mat. Fundam. Napr., 39, 36–65 (2011).Google Scholar

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

Authors and Affiliations

  1. 1.M. V. Lomonosov Moscow State UniversityMoscowRussia

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