Skip to main content
SpringerLink
Log in

General decay result of solutions for viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term

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

Abstract

In this paper, we consider a viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term. Under suitable assumptions on relaxation functions, we establish general decay result for energy.

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. Adams, R., Fournier, J.: Sobolev Spaces. Academic Press, New York (2003)

    MATH  Google Scholar 

  2. Apalara, T.A., Messaoudi, S.A., Mustafa, M.I.: Energy decay in thermodlasticity type III with viscoelastic damping and delay term. Electron. J. Differ. Equ. 2012(128), 1–15 (2012)

    MATH  Google Scholar 

  3. Balakrishnan, A.V., Taylor, L.W.: Distributed Parameter Nonlinear Damping Models for Flight Structure, Damping 89. Flight Dynamics Lab and Air Force Wright Aeronautral Labs, WPAFB, Dayton (1989)

    Google Scholar 

  4. Bass, R.W., Zes, D.: Spillover nonlinearity and flexible structures. In: Taylor, L.W. (ed.), The Fourth NASA Workshop Computational Control of Flexible Aerospace Systems, NASA ConFlight Dynamic Lab and Air Force Wright Aeronautral Labs, WPAFB (1989), Conference Publication 10065, 1991, pp. 1–14

  5. Dai, Q.Y., Yang, Z.F.: Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay. Z. Angew. Math. Phys. 65(5), 885–903 (2014)

    Article  MathSciNet  Google Scholar 

  6. Datko, R.: Not all feedback stabilized hyperbolic systems are robust with respect to small time delay in their feedbacks. SIAM J. Control Optim. 26(3), 697–713 (1988)

    Article  MathSciNet  Google Scholar 

  7. Feng, B.: Well-posedness and exponential stability for a plate equation with time-varying delay and past history. Z. Angew. Math. Phys. 68, 6 (2017)

    Article  MathSciNet  Google Scholar 

  8. Feng, B.: Global well-posedness and stability for a viscoelastic plate equation with a time delay. Math. Probl. Eng. 2015, 1–10 (2015)

    MathSciNet  Google Scholar 

  9. Feng, B., Soufyane, A.: Existence and decay rates for a coupled Balakrishnan–Taylor viscoelastic system with dynamic boundary conditions. Math. Methods Appl. Sci. 43, 3375–3391 (2020)

    Article  Google Scholar 

  10. Ha, T.G.: General decay rate estimates for viscoelastic wave equation with Balakrishnan–Taylor damping. Z. Angew. Math. Phys. 67, 32 (2016)

    Article  MathSciNet  Google Scholar 

  11. Hao, J.H., Hou, Y.X.: Stabilization for wave equation of variable coefficients with Balakrishnan–Taylor damping and source term. Comput. Math. Appl. 76, 2235–2245 (2018)

    Article  MathSciNet  Google Scholar 

  12. Hao, J.H., Wang, F.: General decay rate for weak viscoelastic wave equation with Balakrishnan–Taylor damping and time-varying delay. Comput. Math. Appl. 334, 168–173 (2018)

    Article  Google Scholar 

  13. Hao, J.H., Wang, P.P.: General decay result for thermoelastic beam equation system with time-varying delay. Appl. Math. Comput. 334, 168–173 (2018)

    Article  MathSciNet  Google Scholar 

  14. Kang, Y.H., Lee, M.J., Park, J.Y.: Asymptotic stability of a viscoelastic problem with Balakrishnan–Taylor damping and time-varying delay. Comput. Math. Appl. 74, 1506–1515 (2017)

    Article  MathSciNet  Google Scholar 

  15. Kirane, M., Said-Houari, B.: Existence and asymptotic stability of a viscoelastic wave equation with a delay. Z. Angew. Math. Phys. 62, 1065–1082 (2011)

    Article  MathSciNet  Google Scholar 

  16. Lee, M.J., Park, J.Y., Kang, Y.H.: Asymptotic stability of a problem with Balakrishnan–Taylor damping and a time delay. Comput. Math. Appl. 70, 478–487 (2015)

    Article  MathSciNet  Google Scholar 

  17. Liu, G.W., Zhang, H.W.: Well-posedness for a class of wave equation with past history and a delay. Z. Angew. Math. Phys. 7(1), 1–14 (2003)

    MathSciNet  Google Scholar 

  18. Li, G., Wang, D.H., Zhu, B.Q.: Well-posedness and general decay of solution for a transmission problem with past history and delay. Electron. J. Differ. Equ. 2016(23), 1–21 (2016)

    Google Scholar 

  19. Liu, W., Zhu, B., Li, G., Wang, D.: General decay for a viscoelastic Kirchhoff equation with Balakrishnan–Taylor damping, dynamic boundary conditions and a time-varying delay term. Evol. Equ. Control Theory 6, 239–260 (2017)

    Article  MathSciNet  Google Scholar 

  20. Mu, C.L., Ma, J.: On a system of nonlinear wave equations with Balakrishnan–Taylor damping. Z. Angew. Math. Phys. 65, 91–113 (2014)

    Article  MathSciNet  Google Scholar 

  21. Nicaise, S., Pignotti, C.: Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45, 1561–1585 (2006)

    Article  MathSciNet  Google Scholar 

  22. Nicaise, S., Pignotti, C.: Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 21, 935–958 (2008)

    MathSciNet  MATH  Google Scholar 

  23. Nicaise, S., Pignotti, C.: Interior feedback stabilization of wave equations with time dependence delay. Electron. J. Differ. Equ. 41, 1–20 (2011)

    MATH  Google Scholar 

  24. Park, S.H.: Decay rate estimates for a weak viscoelastic beam equation with time-varying delay. Appl. Math. Lett. 31, 46–51 (2014)

    Article  MathSciNet  Google Scholar 

  25. Tatar, N.-E., Zarai, A.: Exponential stability and blow up for a problem with Balakrishnan–Taylor damping. Demonstr. Math. XLIV 1, 67–90 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Tatar, N.-E., Zaran, A.: On a Kirchhoff equation with Balakrishnan–Taylor damping and source term. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 18(5), 615–627 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Wu, S.T.: General decay of solutions for a viscoelastic equtaion with Balakrishnan–Talyor dampimg. Taiwan. J. Math. 19, 553–566 (2015)

    Article  Google Scholar 

  28. You, Y.: Inertial manifolds and stabilization of nonlinear beam equations with Balakrishnan–Taylor damping. Abstr. Appl. Anal. 1(1), 83–102 (1996)

    Article  MathSciNet  Google Scholar 

  29. Zarai, A., Tatar, N.-E.: Global existence and polynomial decay for a problem with Balakrishnan–Taylor damping. Arch. Math. (BRNO) 46, 157–176 (2010)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the Directorate-General for Scientific Research and Technological Development (DGRSDT), Algeria.

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Larbi Ben M’Hidi University, 04000, Oum El-Bouaghi, Algeria

    Billel Gheraibia

  2. Department of Mathematics and Computer Science, Larbi Tebessi University, 12000, Tebessa, Algeria

    Nouri Boumaza

  3. Laboratory of Mathematics, Informatics and Systems (LAMIS), Larbi Tebessi University, Tebessa, Algeria

    Billel Gheraibia & Nouri Boumaza

Authors
  1. Billel Gheraibia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nouri Boumaza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nouri Boumaza.

Additional information

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

Gheraibia, B., Boumaza, N. General decay result of solutions for viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term. Z. Angew. Math. Phys. 71, 198 (2020). https://doi.org/10.1007/s00033-020-01426-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-020-01426-1

Keywords

Mathematics Subject Classification

Search

Navigation