Advertisement

Journal of Mathematical Sciences

, Volume 244, Issue 2, pp 170–182 | Cite as

A Study of Operator Models Arising in Problems of Hereditary Mechanics

  • V. V. Vlasov
  • N. A. RautianEmail author
Article
  • 4 Downloads

Abstract

We examine integro-differential equations with unbounded operator-valued coefficients. The principal part of such an equation is an abstract hyperbolic operator perturbed by Volterra integral operators whose kernels are fractional exponential functions of the type occurring in viscoelasticity.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Yu. N. Rabotnov, Elements of Hereditary Solid Mechanics, Mir, Moscow (1980).zbMATHGoogle Scholar
  2. 2.
    A. A. Ilyushin and B. E. Pobedrya, Principles of the Mathematical Theory of Thermoviscoelasticity [in Russian], Nauka, Moscow (1970).Google Scholar
  3. 3.
    M. E. Gurtin and A. C. Pipkin, “General theory of heat conduction with finite wave speed,” Arch. Rat. Mech. Anal., 31, 113–126 (1968).MathSciNetCrossRefGoogle Scholar
  4. 4.
    A. Eremenko and S. Ivanov, “Spectra of the Gurtin–Pipkin type equations,” SIAM J. Math. Anal., 43, No. 5, 2296–2306 (2011).MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. V. Lykov, The Problem of Heat and Mass Transfer [in Russian], Nauka i Tekhnika, Minsk (1976).Google Scholar
  6. 6.
    V. V. Vlasov, A. A. Gavrikov, S. A. Ivanov, D. Yu. Knyaz’kov, V. A. Samarin, and A. S. Shamaev, “Spectral properties of combined media,” J. Math. Sci., 164, No. 6, 948–963 (2010).MathSciNetCrossRefGoogle Scholar
  7. 7.
    V. V. Zhikov, “On an extension of the method of two-scale convergence and its applications,” Mat. Sb., 191, No. 7, 31–72 (2000).MathSciNetCrossRefGoogle Scholar
  8. 8.
    V. V. Vlasov and N. A. Rautian, “Well-defined solvability and spectral analysis of abstract hyperbolic integrodifferential equations,” Tr. Semin. Petrovskogo, 28, 75–114 (2011).MathSciNetzbMATHGoogle Scholar
  9. 9.
    V. V. Vlasov, N. A. Rautian, and A. S. Shamaev, “Solvability and spectral analysis of integrodifferential equations arising in thermal physics and acoustics,” Dokl. RAN, 434, No. 1, 12–15 (2010).Google Scholar
  10. 10.
    V. V. Vlasov, N. A. Rautian, and A. S. Shamaev, “Spectral analysis and well-defined solvability of abstract integro-differential equations arising in thermal physics and acoustics,” Sovrem. Mat. Fund. Napr., 39, 36–65 (2011).Google Scholar
  11. 11.
    V. V. Vlasov and N. A. Rautian, “Well-defined solvability and spectral analysis of integro-differential equations arising in the theory of viscoelasticity,” Sovrem. Mat. Fund. Napr., 58, 22–42 (2015).Google Scholar
  12. 12.
    V. V. Vlasov and N. A. Rautian, “Well-defined solvability of Volterra differential equations in Hilbert space,” Tr. Mosk. Mat. Obshch., 75, No. 2, 131–155 (2014).Google Scholar
  13. 13.
    V. V. Vlasov and N. A. Rautian, Spectral Analysis of Functional-Differential Equations [in Russian], MAKS Press, Moscow (2016).zbMATHGoogle Scholar
  14. 14.
    J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, Berlin (1972).CrossRefGoogle Scholar
  15. 15.
    H. Bateman and A. Erdelyi, Tables of Integral Transforms, Vols. 1 and 2, McGraw-Hill (1954).Google Scholar
  16. 16.
    K. Yosida, Functional Analysis, Springer, Berlin (1995).CrossRefGoogle Scholar
  17. 17.
    R. Perez Ortiz, V. V. Vlasov, N. A. Rautian, “Spectral analysis of Volterra integrodifferential equations with the kernels depending on a parameter,” arxiv.org/abs/1710.07112 (2017).
  18. 18.
    B. Sz Nagy, C. Foias, H. Bercovici, and L. Kérchy, Harmonic Analysis of Operators on Hilbert Space, Springer, Berlin (2010).CrossRefGoogle Scholar
  19. 19.
    V. V. Vlasov, R. Perez Ortiz, and N. A. Rautian, “Study of Volterra integro-differential equations with kernels depending on a parameter,” Differ. Equ., 54, No. 3, 363–380 (2018).MathSciNetCrossRefGoogle Scholar
  20. 20.
    R. Perez Ortiz and V. V. Vlasov, “Correct solvability of volterra integrodifferential equations in Hilbert space,” Electron. J. Qualit. Theory Differ. Equ., 31, 1–17 (2016).MathSciNetzbMATHGoogle Scholar
  21. 21.
    L. Pandolfi, “The controllability of the Gurtin–Pipkin equations: A cosine operator approach,” Appl. Math. Optim., 52, 143–165 (2005).MathSciNetCrossRefGoogle Scholar
  22. 22.
    A. L. Skubachevskii, “A class of functional-differential operators satisfying the Kato hypothesis,” Algebra Anal., 30, No. 2, 249–273 (2018).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

Personalised recommendations