Skip to main content
SpringerLink
Log in

Exponential Stability of a Certain Semigroup and Applications

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

The uniform exponential stability of a C0-semigroup with generator of a special form is proved. Such semigroups arise in the study of various problems of the theory of viscoelasticity. The proved statement is applied to the study of the asymptotic behavior of solutions in the problem of small motions of a viscoelastic body subject to driving forces of a special form.

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

    Article  MathSciNet  MATH  Google Scholar 

  2. C. M. Dafermos, “On abstract Volterra equations with applications to linear viscoelasticity,” J. Differential Equations 7, 554–569 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Renardy, W. J. Hrusa, and J. A. Nohel, Mathematical Problems in Viscoelasticity, in PitmanMonogr. Surveys Pure Appl. Math. (Longman Sci. and Tec., Harlow, 1987), Vol.35.

  4. M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, in SIAM Stud. Appl.Math. (SIAM, Philadelphia, PA, 1992), Vol.12.

  5. Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, in Chapman & Hall/CRC Res. NotesMath. (Chapman & Hall/CRC, Boca Raton, FL, 1999), Vol.398.

  6. J. Lagnese, “Decay of solutions of wave equations in a bounded region with boundary dissipation,” J. Differential Equations 50 (2), 163–182 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Fabrizio and B. Lazzari, “On the existence and the asymptotic stability of solutions for linearly viscoelastic solids,” Arch. Rational Mech. Anal. 116 (2), 139–152 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  8. J. E. Mu noz Rivera, “Asymptotic behaviour in linear viscoelasticity,” Quart. Appl. Math. 52 (4), 629–648 (1994).

    MathSciNet  MATH  Google Scholar 

  9. W. Liu, “The exponential stabilization of the higher-dimensional linear systemofrmoviscoelasticity,” J.Math. Pures Appl. (9) 77 (4), 355–386 (1998).

    Article  MathSciNet  Google Scholar 

  10. F. Alsobau-Boussouria, J. Prüss, and R. Zacher, “Exponential and polynomial stability of a wave equation for boundary memory damping with singular kernels,” C. R. Math. Acad. Sci. Paris 347, 277–282 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  11. V. V. Vlasov and N. A. Rautian, “Well-defined solvability and spectral analysis of integro-differential equations appearing in the theory of viscoelasticity,” in Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22–29, 2014), Part 1 (RUDN, Moscow, 2015), Vol. 58, pp. 22–42 [in Russian].

    Google Scholar 

  12. J. A. D. Apple M. Fabrizio, and B. Lazzari, “On exponential asymptotic stability in linear viscoelasticity,” Math. ModelsMethods Appl. Sci. 16 (10), 1677–1694 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  13. J. E. Mu noz Rivera and M. G. Naso, “On the decay of the energy for systems with memory and indefinite dissipation,” Asymptot. Anal. 49, 189–204 (2006).

    MathSciNet  MATH  Google Scholar 

  14. J. E. Mu noz Rivera and M. G. Naso, “Asymptotic stability of semigroups associated with linear weak dissipative systems with memory,” J. Math. Anal. Appl. 326, 691–707 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  15. F. Alsobau-Boussouria, P. Cannarsa, and D. Sforza, “Decay estimates for second-order evolution equations with memory,” J. Funct. Anal. 254 (5), 1342–1372 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Alsobau-Boussouria and P. Cannarsa, “A general method for proving sharp energy decay rates for memory-dissipative evolution equations,” C. R. Math. Acad. Sci. Paris 347, 867–872 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  17. A. G. Baskakov, “Estimates for the Green’s function and parameters of exponential dichotomy of a hyperbolic operator semigroup and linear relations,” Mat. Sb. 206 (8), 23–62 (2015) [Sb. Math. 206 (8), 1049–1086 (2015)].

    Article  MathSciNet  MATH  Google Scholar 

  18. A. B. Muravnik, “Asymptotic properties of solutions of the Dirichlet problem in the half-plane for differential-difference elliptic equations,” Mat. Zametki 100 (4), 566–576 (2016) [Math. Notes 100 (4), 579–588 (2016)].

    Article  MathSciNet  MATH  Google Scholar 

  19. A. V. Vestyak and O. A. Matevosyan, “Behavior of the solution of the Cauchy problem for a hyperbolic equation with periodic coefficients,” Mat. Zametki 100 (5), 766–769 (2016) [Math. Notes 100 (5), 751–755 (2016)].

    Article  MathSciNet  MATH  Google Scholar 

  20. A. G. Baskakov and V. D. Kharitonov, “Spectral analysis of operator polynomials and higher-order difference operators,” Mat. Zametki 101 (3), 330–345 (2017) [Math. Notes 101 (3), 391–405 (2017)].

    Article  MathSciNet  MATH  Google Scholar 

  21. R. O. Griniv and A. A. Shkalikov, “Exponential stability of semigroups related to operator models in mechanics,” Mat. Zametki 73 (5), 657–664 (2003) [Math. Notes 73 (5), 618–624 (2003)].

    Article  MathSciNet  MATH  Google Scholar 

  22. L. Gearhart, “Spectral theory for contraction semigroups on Hilbert spaces,” Trans. Amer. Math. Soc. 236, 385–394 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  23. K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, in Grad. Texts in Math. (Springer-Verlag, Berlin, 2000), Vol.194.

  24. T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Heidelberg, 1966; Mir, Moscow, 1972).

    Google Scholar 

  25. M. L. Heard, “An abstract semilinear hyperbolic Volterra integro-differential equation,” J. Math. Anal. Appl. 80, 175–202 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  26. W. Desch, R. Grimmer, and W. Schappacher, “Some considerations for linear integro-differential equations,” J. Math. Anal. Appl. 104, 219–234 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  27. N. D. Kopachevsky and S. G. Krein, Operator Approach to Linear Problems of Hydrodynamics. Vol. 2. Nonself-Adjoint Problems for Viscous Fluids, in Oper. Theory Adv. Appl. (Birkhäuser Verlag, Basel, 2003), Vol.146.

  28. S. G. Krein, Linear Differential Equations in a Banach Space (Nauka, Moscow, 1967) [in Russian].

    Google Scholar 

  29. A. A. Il’yushin and B. E. Pobedrya, Foundations of the Mathematical Theory of Thermoviscoelasticity (Nauka, Moscow, 1970) [in Russian].

    Google Scholar 

  30. K. Rektorys, Variational Methods in Mathematics, Science, and Engineering (Reidel, Dordrecht, 1983; Mir, Moscow, 1985).

    MATH  Google Scholar 

  31. N. D. Kopachevsky and S. G. Krein, in Oper. Theory Adv. Appl., Vol. 128: Operator Approach to Linear Problems of Hydrodynamics. Vol. 1: Self-Adjoint Problems for an Ideal Fluid (Birkhäuser Verlag, Basel, 2001).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Voronezh State University, Voronezh, Russia

    D. A. Zakora

Authors
  1. D. A. Zakora
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. A. Zakora.

Additional information

Original Russian Text © D. A. Zakora, 2018, published in Matematicheskie Zametki, 2018, Vol. 103, No. 5, pp. 702–719.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zakora, D.A. Exponential Stability of a Certain Semigroup and Applications. Math Notes 103, 745–760 (2018). https://doi.org/10.1134/S0001434618050073

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434618050073

Keywords

Search

Navigation