Skip to main content
SpringerLink
Log in

Stabilization for the Wave Equation with Singular Kelvin–Voigt Damping

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We consider the wave equation with Kelvin–Voigt damping in a bounded domain. The exponential stability result proposed by Liu and Rao (Z Angew Math Phys (ZAMP) 57:419–432, 2006) or Tebou (C R Acad Sci Paris Ser I 350: 603–608, 2012) for that system assumes that the damping is localized in a neighborhood of the whole or a part of the boundary under some consideration. In this paper we propose to deal with this geometrical condition by considering a singular Kelvin–Voigt damping which is localized far away from the boundary. In this particular case Liu and Liu (SIAM J Control Optim 36:1086–1098, 1998) proved the lack of the uniform decay of the energy. However, we show that the energy of the wave equation decreases logarithmically to zero as time goes to infinity. Our method is based on the frequency domain method. The main feature of our contribution is to write the resolvent problem as a transmission system to which we apply a specific Carleman estimate.

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.

Fig. 1

Similar content being viewed by others

References

  1. Ammari , K., Nicaise , S.: Stabilization of Elastic Systems by Collocated Feedback, 2124. Springer, Cham 2015

    MATH  Google Scholar 

  2. Banks , H.T., Smith , R.C., Wang , Y.: Modeling aspects for piezoelectric patch actuation of shells, plates and beams. Quart. Appl. Math. LIII, 353–381, 1995

    Article  Google Scholar 

  3. Bardos , C., Lebeau , G., Rauch , J.: Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30, 1024–1065, 1992

    Article  MathSciNet  Google Scholar 

  4. Batty , C.J.K., Duyckaerts , T.: Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equ. 8, 765–780, 2008

    Article  MathSciNet  Google Scholar 

  5. Bellasoued , M.: Carleman estimates and distribution of resonnances for the transparent obstacle and application to the stabilization. Asymptot. Anal. 35, 257–279, 2003

    MathSciNet  Google Scholar 

  6. Borichev , A., Tomilov , Y.: Optimal polynomial decay of function and operator semigroups. Math. Ann. 347(2), 455–478, 2010

    Article  MathSciNet  Google Scholar 

  7. Burq , N.: Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel. Acta Math. 180, 1–29, 1998

    Article  MathSciNet  Google Scholar 

  8. Chen , S., Liu , K., Liu , Z.: Spectrum and stability for elastic systems with global or local Kelvin–Voigt damping. SIAM J. Appl. Math. 59, 651–668, 1999

    Article  MathSciNet  Google Scholar 

  9. Duyckaerts , T.: Optimal decay rates of the energy of a hyperbolic–parabolic system coupled by an interface. Asymptot. Anal. 51, 17–45, 2007

    MathSciNet  MATH  Google Scholar 

  10. Hassine , F.: Stability of elastic transmission systems with a local Kelvin-Voigt damping. Eur. J. Control23, 84–93, 2015

    Article  MathSciNet  Google Scholar 

  11. Hassine , F.: Asymptotic behavior of the transmission Euler–Bernoulli plate and wave equation with a localized Kelvin–Voigt damping. Discrete Cont. Dyn. Syst. Ser. B21, 1757–1774, 2016

    Article  MathSciNet  Google Scholar 

  12. Hassine , F.: Energy decay estimates of elastic transmission wave/beam systems with a local Kelvin–Voigt damping. Int. J. Control89(10), 1933–1950, 2016

    Article  MathSciNet  Google Scholar 

  13. Hassine , F.: Logarithmic stabilization of the Euler–Bernoulli plate equation with locally distributed Kelvin–Voigt damping. Evol. Equ. Control Theory455(2), 1765–1782, 2017

    MathSciNet  MATH  Google Scholar 

  14. Lebeau, G.: Équations des ondes amorties, Algebraic and geometric methods in mathematical physics. Math. Phys. Stud., vol. 19 (Kaciveli, 1993), Kluwer Acad. Publ., Dordrecht, 73–109, 1996

  15. Lebeau , G., Robbiano , L.: Stabilisation de l’équation des ondes par le bord. Duke Math. J. 86, 465–491, 1997

    Article  MathSciNet  Google Scholar 

  16. Liu , K., Liu , Z.: Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin–Voigt damping. SIAM J. Control Optim. 36, 1086–1098, 1998

    Article  MathSciNet  Google Scholar 

  17. Liu , K.S., Rao , B.: Characterization of polynomial decay rate for the solution of linear evolution equation. Z. Angew. Math. Phys. (ZAMP)56, 630–644, 2005

    Article  MathSciNet  Google Scholar 

  18. Liu , K.S., Rao , B.: Exponential stability for wave equations with local Kelvin–Voigt damping. Z. Angew. Math. Phys. (ZAMP)57, 419–432, 2006

    Article  MathSciNet  Google Scholar 

  19. Pazy , A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York 1983

    Book  Google Scholar 

  20. Le Rousseau , J., Robbiano , L.: Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations. Arch. Rational Mech. Anal. 195, 953–990, 2010

    Article  ADS  MathSciNet  Google Scholar 

  21. Tebou, L.: A constructive method for the stabilization of the wave equation with localized Kelvin–Voigt damping. C. R. Acad. Sci. Paris Ser. I350, 603–608, 2012

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the referees for their valuable comments which enabled us to improve the paper.

Author information

Authors and Affiliations

  1. UR Analysis and Control of PDE, UR 13ES64, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019, Monastir, Tunisia

    Kaïs Ammari & Fathi Hassine

  2. Laboratoire de Mathématiques, Université de Versailles Saint-Quentin en Yvelines, 78035, Versailles, France

    Luc Robbiano

Authors
  1. Kaïs Ammari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fathi Hassine
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luc Robbiano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kaïs Ammari.

Additional information

Communicated by P. Constantin

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ammari, K., Hassine, F. & Robbiano, L. Stabilization for the Wave Equation with Singular Kelvin–Voigt Damping. Arch Rational Mech Anal 236, 577–601 (2020). https://doi.org/10.1007/s00205-019-01476-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-019-01476-4

Search

Navigation