Skip to main content
SpringerLink
Log in

On the Properties of Semigroups Generated by Volterra Integro-Differential Equations

  • SHORT COMMUNICATIONS
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We study abstract integro-differential equations serving as operator models for problems arising in viscoelasticity. The results are based on an approach related to the study of one-parameter semigroups for linear evolution equations. The approach proposed can be used when investigating other integro-differential equations containing integral terms of the form of Volterra convolution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. Kato, T., Perturbation theory for linear operators, Berlin–Heidelberg: Springer-Verlag, 1966. Translated under the title:Teoriya vozmushchenii lineinykh operatorov, Moscow: Mir, 1972.

    Book  Google Scholar 

  2. Il’yushin, A.A. and Pobedrya, B.E., Osnovy matematicheskoi teorii termovyazkouprugosti (Basics of Mathematical Theory of Thermoviscoelasticity), Moscow: Nauka, 1970.

    Google Scholar 

  3. Christensen, R.M., Theory of Viscoelasticity. An Introduction, New York–London: Academic Press, 1971.

    Google Scholar 

  4. Gurtin, M.E. and Pipkin, A.C., General theory of heat conduction with finite wave speed, Arch. Rat. Mech. Anal., 1968, vol. 31, pp. 113–126.

    Article  MathSciNet  Google Scholar 

  5. Lykov, A.V., Problema teplo- i massoobmena (Problems of Heat and Mass Exchange), Minsk: Nauka Tekh., 1976.

    Google Scholar 

  6. Rabotnov, Yu.N., Elementy nasledstvennoi mekhaniki tverdykh tel (Elements of Hereditary Mechanics of Solids), Moscow: Nauka, 1977.

    Google Scholar 

  7. Vlasov, V.V. and Rautian, N.A., Spektral’nyi analiz funktsional’no-differentsial’nykh uravnenii (Spectral Analysis of Functional-Differential Equations), Moscow: MAKS PRess, 2016.

    Google Scholar 

  8. Vlasov, V.V. and Rautian, N.A., Well-posedness and spectral analysis of integrodifferential equations arising in viscoelasticity, Theory J. Math. Sci., 2018, vol. 233, no. 4, pp. 555–577.

    Article  Google Scholar 

  9. Vlasov, V.V. and Rautian, N.A., Well-posed solvability and the representation of solutions of integro-differential equations arising in viscoelasticity, Differ. Equations, 2019, vol. 55, no. 4, pp. 561–574.

    Article  MathSciNet  Google Scholar 

  10. Vlasov, V.V. and Rautian, N.A., A study of operator models arising in problems of hereditary mechanics, J. Math. Sci., 2020, vol. 244, no. 2, pp. 170–182.

    Article  MathSciNet  Google Scholar 

  11. Eremenko, A. and Ivanov, S., Spectra of the Gurtin–Pipkin type equations,SIAM J. Math. Anal., 2011, vol. 43, no. 5, pp. 2296–2306.

    Article  MathSciNet  Google Scholar 

  12. Engel, K.J. and Nagel, R., One-Parameter Semigroups for Linear Evolution Equations, New York: Springer, 2000.

    MATH  Google Scholar 

  13. Amendola, G., Fabrizio, M., and Golden, J.M., Thermodynamics of Materials with Memory. Theory and Applications, New York–Dordrecht–Heidelberg–London: Springer, 2012.

    Book  Google Scholar 

Download references

Funding

This work was financially supported by the Russian Foundation for Basic Research, project no. 20-01-00288A.

Author information

Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    V. V. Vlasov & N. A. Rautian

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991, Russia

    V. V. Vlasov & N. A. Rautian

Authors
  1. V. V. Vlasov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. A. Rautian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to V. V. Vlasov or N. A. Rautian.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vlasov, V.V., Rautian, N.A. On the Properties of Semigroups Generated by Volterra Integro-Differential Equations. Diff Equat 56, 1100–1104 (2020). https://doi.org/10.1134/S001226612008011X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S001226612008011X

Search

Navigation