Skip to main content
SpringerLink
Log in

Uniform and weak stability of Bresse system with two infinite memories

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this paper, we consider one-dimensional linear Bresse systems in a bounded open domain under Dirichlet–Neumann–Neumann boundary conditions with two infinite memories acting only on two equations. First, we establish the well-posedness in the sense of semigroup theory. Then, we prove two (uniform and weak) decay estimates depending on the speeds of wave propagations, the smoothness of initial data and the arbitrarily growth at infinity of the two relaxation functions.

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. Alabau-Boussouira F., Muñoz Rivera J.E., Almeida Júnior D.S.: Stability to weak dissipative Bresse system. J. Math. Anal. Appl. 374, 481–498 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alves M.O., Fatori L.H., Jorge Silva M.A., Monteiro R.N.: Stability and optimality of decay rate for weakly dissipative Bresse system. Math. Meth. Appl. Sci. 38, 898–908 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bresse J.A.C.: Cours de Méchanique Appliquée. Mallet Bachelier, Paris (1859)

    MATH  Google Scholar 

  4. Charles W., Soriano J.A., Nascimentoc F.A., Rodrigues J.H.: Decay rates for Bresse system with arbitrary nonlinear localized damping. J. Diff. Equa. 255, 2267–2290 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. De Lima Santos M., Soufyane A., Da Silva Almeida Júnior D.: Asymptotic behavior to Bresse system with past history. Quart. Appl. Math. 73, 23–54 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fatori L.H., Monteiro R.N.: The optimal decay rate for a weak dissipative Bresse system. Appl. Math. Lett. 25, 600–604 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fatori L.H., Muñoz Rivera J.E.: Rates of decay to weak thermoelastic Bresse system. IMA J. Appl. Math. 75, 881–904 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fernández Sare H.D., Muñoz Rivera J.E.: Stability of Timoshenko systems with past history. J. Math. Anal. Appl. 339, 482–502 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guesmia A.: Asymptotic stability of abstract dissipative systems with infinite memory. J. Math. Anal. Appl. 382, 748–760 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guesmia A.: Asymptotic behavior for coupled abstract evolution equations with one infinite memory. Applicable Analysis 94, 184–217 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Guesmia A., Kafini M.: Bresse system with infinite memories. Math. Meth. Appl. Sci. 38, 2389–2402 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Komornik V.: Exact Controllability and Stabilization. The Multiplier Method. Masson-John Wiley, Paris (1994)

    MATH  Google Scholar 

  14. Lagnese J.E., Leugering G., Schmidt J.P.: Modelling of dynamic networks of thin thermoelastic beams. Math. Meth. Appl. Scie. 16, 327–358 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lagnese, J.E., Leugering, G., Schmidt, E.J.P.G.: Modelling, Analysis and Control of Dynamic Elastic Multi-Link Structures. Systems & Control: Foundations & Applications. Birkhäuser, Boston. ISBN: 0-8176-3705-2 (1994)

  16. Liu Z., Rao B.: Energy decay rate of the thermoelastic Bresse system. Z. Angew. Math. Phys. 60, 54–69 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Najdi N., Wehbe A.: Weakly locally thermal stabilization of Bresse systems. Elec. J. Diff. Equa. 2014, 1–19 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Noun N., Wehbe A.: Weakly locally internal stabilization of elastic Bresse system. C. R. Acad. Scie. Paris Sér. I 350, 493–498 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pazy A.: Semigroups of linear operators and applications to partial differential equations. Springer, New York (1983)

    Book  MATH  Google Scholar 

  20. Soriano J.A., Charles W., Schulz R.: Asymptotic stability for Bresse systems. J. Math. Anal. Appl. 412, 369–380 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Soriano J.A., Muñoz Rivera J.E., Fatori L.H.: Bresse system with indefinite damping. J. Math. Anal. Appl. 387, 284–290 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Soufyane A., Said-Houari B.: The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system. Evol. Equa. Cont. Theory 3, 713–738 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wehbe A., Youssef W.: Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks. J. Math. Phys. 51, 1–17 (2010)

    MathSciNet  MATH  Google Scholar 

  24. Youssef, W.: Contrôle et stabilisation de systèmes élastiques couplés, PhD thesis, University of Metz, France (2009)

Download references

Author information

Authors and Affiliations

  1. Institut Elie Cartan de Lorraine, UMR 7502, Université de Lorraine, Bat. A, Ile du Saulcy, 57045, Metz Cedex 01, France

    Aissa Guesmia

  2. LaSIE, Faculté des Sciences, Pôle Sciences et Technologies, Université de la Rochelle, La Rochelle, France

    Mokhtar Kirane

Authors
  1. Aissa Guesmia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mokhtar Kirane
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aissa Guesmia.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guesmia, A., Kirane, M. Uniform and weak stability of Bresse system with two infinite memories. Z. Angew. Math. Phys. 67, 124 (2016). https://doi.org/10.1007/s00033-016-0719-y

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-016-0719-y

Mathematics Subject Classification

Keywords

Search

Navigation