Skip to main content
SpringerLink
Log in

Rate of decay of some abstract Petrowsky-like dissipative semi-groups

  • RESEARCH ARTICLE
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

In this paper, we show that the fastest decay rate for some Petrowsky-like dissipative semi-groups is given by the supremum of the real part of the spectrum of the associated infinitesimal generator, if the corresponding operator satisfies some spectral gap condition. We give also some applications to illustrate our setting.

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

Fig. 1

Similar content being viewed by others

References

  1. Ammari, K., Henrot, A., Tucsnak, M.: Optimal location of the actuator for the pointwise stabilization of a string. C. R. Acad. Sci. Paris Sér. I Math. 330(4), 275–280 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ammari, K., Henrot, A., Tucsnak, M.: Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string. Asymptot. Anal. 28(3–4), 215–240 (2001)

    MathSciNet  MATH  Google Scholar 

  3. Ammari, K., Tucsnak, M.: Stabilization of second order evolution equations by a class of unbounded feedback. ESAIM Control Optim. Calc. Var. 6, 361–386 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ammari, K., Dimassi, M., Zerzeri, M.: The rate at which energy decays in a viscously damped hinged Euler–Bernoulli beam. J. Differ. Equ. 257(9), 3501–3520 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benaddi, A., Rao, B.: Energy decay rate of wave equations with indefinite damping. J. Differ. Equ. 161(2), 337–357 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Castro, C., Cox, S.: Achieving arbitrarily large decay in the damped wave equation. SIAM J. Control Optim. 39(6), 1748–1755 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cox, S., Zuazua, E.: The rate at which energy decays in a damped string. Commun. Partial Differ. Equ. 19(1–2), 213–243 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cox, S., Zuazua, E.: The rate at which energy decays in a string damped at one end. Indiana Univ. Math. J. 44(2), 545–573 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. Freitas, P.: Optimizing the rate of decay of solutions of the wave equation using genetic algorithms: a counterexample to the constant damping conjecture. SIAM J. Control Optim. 37(2), 376–387 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gohberg, I.C., Kreĭn, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators, vol. 18. American Mathematical Society, Providence, RI (1969)

    MATH  Google Scholar 

  11. Haraux, A.: Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portugal. Math. 46(3), 245–258 (1989)

    MathSciNet  MATH  Google Scholar 

  12. Kato, T.: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1966)

    Book  MATH  Google Scholar 

  13. Lebeau, G.: Équation des ondes amorties. Algebraic and geometric methods in mathematical physics (kaciveli, 1993). Math. Phys. Stud. 19, 73–109 (1996)

    MathSciNet  Google Scholar 

  14. Pöschel, J., Trubowitz, E.: Inverse Spectral Theory, Pure and Applied Mathematics, vol. 130. Academic Press Inc, Boston, MA (1987)

    MATH  Google Scholar 

  15. Ramm, A.-G.: On the basis property for root vectors of some nonselfadjoint operators. J. Math. Anal. Appl. 80, 57–66 (1981)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the referee for his thorough and careful reading of the paper.

Author information

Authors and Affiliations

  1. UR Analyse et Contrôle des Edp, UR 13ES64, Dép. de Mathématiques, Faculté des Sciences de Monastir, Université de Monastir, 5019, Monastir, Tunisie

    Kaïs Ammari

  2. Université Bordeaux 1, CNRS, UMR 5251 IMB, 351, Cours de la Libration, 33405, Talence Cedex, France

    Mouez Dimassi

  3. Université Paris XIII, CNRS, UMR 7539 LAGA, 99, Ave Jean-Baptiste Clément, 93430, Villetaneuse, France

    Maher Zerzeri

Authors
  1. Kaïs Ammari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mouez Dimassi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Maher Zerzeri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maher Zerzeri.

Additional information

Communicated by Abdelaziz Rhandi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ammari, K., Dimassi, M. & Zerzeri, M. Rate of decay of some abstract Petrowsky-like dissipative semi-groups. Semigroup Forum 93, 1–16 (2016). https://doi.org/10.1007/s00233-015-9728-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-015-9728-y

Keywords

Search

Navigation