Skip to main content
Log in

Forced Motions of Thermoelastic Systems of Memory Type

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we consider the hyperbolic thermoelastic system of memory type and the corresponding system of abstract integro-differential equations. We study forced motions of this system of abstract integro-differential equations and investigate the asymptotic behavior of solutions to this system as time goes to infinity.

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

Access this article

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. B. D. Coleman and M. E. Gurtin, ‘‘Equipresence and constitutive equations for rigid heat conductors,’’ Z. Angew. Math. Phys. 18, 199–208 (1967).

    Article  MathSciNet  Google Scholar 

  2. Ph. Clément and J. A. Nohel, ‘‘Asymptotic behavior of solutions of nonlonear Volterra equations with completely positive kernels,’’ SIAM J. Math. Anal. 12, 514–535 (1981).

    Article  MathSciNet  Google Scholar 

  3. Ph. Clément and J. Prüss, ‘‘Completely positive measures and Feller semigroups,’’ Math. Ann. 287, 73–105 (1990).

    Article  MathSciNet  Google Scholar 

  4. A. Lunardi, ‘‘On the linear heat equation with fading memory,’’ SIAM J. Math. Anal. 21, 1213–1224 (1990).

    Article  MathSciNet  Google Scholar 

  5. M. E. Gurtin and A. C. Pipkin, ‘‘A general theory of heat conduction with finite wave speeds,’’ Arch. Rational Mech. Anal. 31, 113–126 (1968).

    Article  MathSciNet  Google Scholar 

  6. J. Meixner, ‘‘On the linear theory of heat conduction,’’ Arch. Rational Mech. Anal. 39, 108–130 (1971).

    Article  MathSciNet  Google Scholar 

  7. J. W. Nunziato, ‘‘On heat conduction in materials with memory,’’ Quart. Appl. Math. 29, 187–204 (1971).

    Article  MathSciNet  Google Scholar 

  8. V. V. Vlasov and N. A. Rautian, Spectral Analysis of Functional Differential Equations (MAKS Press, Moscow, 2016) [in Russian].

    MATH  Google Scholar 

  9. C. M. Dafermos, ‘‘Asymptotic stability in viscoelasticity,’’ Arch. Rational. Mech. Anal. 37, 297–308 (1970).

    Article  MathSciNet  Google Scholar 

  10. Z. Liu and S. Zheng, ‘‘On the exponential stability of linear viscoelasticity and thermoviscoelasticity,’’ Quart. Appl. Math. 54, 21–31 (1996).

    Article  MathSciNet  Google Scholar 

  11. J. E. Muñoz Rivera, ‘‘Asymptotic behaviour of energy in linear thermoviscoelasticity,’’ Comput. Appl. Math. 11, 45–71 (1992).

    MathSciNet  MATH  Google Scholar 

  12. C. Giorgi and M. G. Naso, ‘‘On the exponential stability of linear non-Fourier thermoviscoelastic bar,’’ Quaderni Sem. Brescia 2/97 (1997).

  13. L. H. Fatori and J. E. Muñoz Rivera, ‘‘Energy decay for hyperbolic thermoelastic systems of memory type,’’ Quart. Appl. Math. 59, 441–458 (2001).

    Article  MathSciNet  Google Scholar 

  14. D. A. Zakora, ‘‘Exponential stability of a certain semigroup and applications,’’ Math. Notes 103, 745–760 (2018).

    Article  MathSciNet  Google Scholar 

  15. K. Rektorys, Variational Methods in Mathematics, Science and Engineering (D. Reidel, Dordrecht, Boston, London, 1980).

    MATH  Google Scholar 

  16. S. G. Krein, Linear Differential Equations in Banach Space, Translations of Mathematical Monographs (Am. Math. Soc., Providence, RI, 1971).

    Google Scholar 

  17. M. Sh. Birman and M. Z. Solomjak, Spectral Theory of Selfadjoint Operators in Hilbert Space (D. Reidel, Dordrecht, Boston, Lancaster, Tokyo, 1987).

    Book  Google Scholar 

  18. O. J. Staffans, Well-Posed Linear Systems (Cambridge Univ. Press, Cambridge, 2005).

    Book  Google Scholar 

  19. Z. Liu and S. Zeng, Semigroups Associated with Dissipative Systems (Chapman and Hall/CRC, Boca Raton, London, New York, Washington, 1999).

    Google Scholar 

  20. L. Gearhart, ‘‘Spectral theory for contraction semigroups on Hilbert spaces,’’ Trans. Am. Math. Soc 236, 385–394 (1978).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to D. A. Zakora.

Additional information

(Submitted by A. B. Muravnik)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zakora, D.A. Forced Motions of Thermoelastic Systems of Memory Type. Lobachevskii J Math 42, 1124–1139 (2021). https://doi.org/10.1134/S199508022105022X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S199508022105022X

Keywords:

Navigation