Skip to main content
SpringerLink
Log in

Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms

  • Published:
Ukrainian Mathematical Journal Aims and scope

The global existence and nonexistence of solutions for a system of nonlinear wave equations with degenerate damping and source terms supplemented with initial and Dirichlet boundary conditions was shown by Rammaha and Sakuntasathien in a bounded domain Ω ⊂ \( {{\mathbb{R}}^n} \), n = 1, 2, 3, in the case where the initial energy is negative. A global nonexistence result on the solution with positive initial energy for a system of viscoelastic wave equations with nonlinear damping and source terms was obtained by Messaoudi and Said-Houari. Our result extends these previous results. We prove that the solutions of a system of wave equations with viscoelastic term, degenerate damping, and strong nonlinear sources acting in both equations at the same time are globally nonexisting provided that the initial data are sufficiently large in a bounded domain Ω of \( {{\mathbb{R}}^n} \), n ≥ 1, the initial energy is positive, and the strongly nonlinear functions f1 and f2 satisfy the appropriate conditions. The main tool of the proof is based on the methods used by Vitillaro and developed by Said-Houari.

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. K. Agre and M. A. Rammaha, “Systems of nonlinear wave equations with damping and source terms,” Different. Integr. Equat., 19, 1235–1270 (2007).

    MathSciNet  Google Scholar 

  2. D. D. Ang and A. P. N. Dinh, “Strong solutions of a quasilinear wave equation with nonlinear damping,” SIAM J. (to appear).

  3. S. Berrimi and S. Messaoudi, “Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping,” Electron. J. Different. Equat., 88, 1–10 (2004).

    MathSciNet  Google Scholar 

  4. S. Berrimi and S. Messaoudi, “Existence and decay of solutions of a viscoelastic equation with a nonlinear source,” Nonlin. Anal., 64, 2314–2331 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  5. M. M. Cavalcanti, D. V. N. Cavalcanti, and J. A. Soriano, “Exponential decay for the solutions of semilinear viscoelastic wave equations with localized damping,” Electron. J. Different. Equat., 44, 1–44 (2002).

    MathSciNet  Google Scholar 

  6. M. M. Cavalcanti and H. P. Oquendo, “Frictional versus viscoelastic damping in a semilinear wave equation,” SIAM J. Contr. Optim., 42, No. 4, 1310–1324 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  7. M. M. Cavalcanti, D. V. N. Cavalcanti, and J. Ferreira, “Existence and uniform decay for nonlinear viscoelastic equation with strong damping,” Math. Meth. Appl. Sci., 24, 1043–1053 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  8. M. M. Cavalcanti, D. V. N. Cavalcanti, P. J. S. Filho, and J. A. Soriano, “Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping,” Different. Integral Equat., 14, No. 1, 85–116 (2001).

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. V. Georgiev and G. Todorova, “Existence of a solution of the wave equation with nonlinear damping and source term,” J. Different. Equat., 109, 295–308 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  11. W. J. Hrusa and M. Renardy, “A model equation for viscoelasticity with a strongly singular kernel,” SIAM. J. Math. Anal., 19, No. 2 (1988).

    Google Scholar 

  12. M. Kafini and S. Messaoudi, “A blow-up result in a Cauchy viscoelastic problem,” Appl. Math. Lett., 21, 549–553 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  13. H. A. Levine and J. Serrin, “Global nonexistence theorems for quasilinear evolution equations with dissipation,” Arch. Ration. Mech. Anal., 37, No. 4, 341–361 (1997).

    Article  MathSciNet  Google Scholar 

  14. H. A. Levine, S. R. Park, and J. Serrin, “Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation,” J. Math. Anal. Appl., 228, No. 1, 181–205 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  15. S. Messaoudi, “Blow up and global existence in a nonlinear viscoelastic wave equation,” Math. Nachr., 260, 58–66 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  16. S. A. Messaoudi, “On the control of solutions of a viscoelastic equation,” J. Franklin Inst., 344, 765–776 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  17. S. A. Messaoudi and B. Said-Houari, “Global nonexistence of solutions of a class of wave equations with nonlinear damping and source terms,” Math. Meth. Appl. Sci., 27, 1687–1696 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  18. S. A. Messaoudi and B. Said-Houari, “Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms,” J. Math. Anal. Appl., 365, 277–287 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Messaoudi and N.-E. Tatar, “Global existence and uniform stability of solutions for quasilinear viscoelastic problem,” Math. Meth. Appl. Sci., 30, 665–680 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Messaoudi and N.-E. Tatar, “Global existence and asymptotic behavior for a nonlinear viscoelastic problem,” Math. Sci. Res., 7, No. 4, 136–149 (2003).

    MathSciNet  MATH  Google Scholar 

  21. Vi. Pata, “Exponential stability in viscoelasticity,” Quart. Appl. Math., 64, No. 3, 499–513 (2006).

    MathSciNet  MATH  Google Scholar 

  22. M. A. Rammaha, Sawanya Sakuntasathien, “Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms,” Nonlin. Anal., 72, 2658–2683 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  23. B. Said-Houari, “Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms,” Different. Integr. Equat., 23, No. 1-2, 79–92 (2010).

    MathSciNet  MATH  Google Scholar 

  24. B. Said-Houari, “Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms,” Z. Angew. Math. Phys., 2011, 62, S. 115–133.

    Article  MathSciNet  MATH  Google Scholar 

  25. B. Said-Houari, S. A. Messaoudi, and A. Guesmia, “General decay of solutions of a nonlinear system of viscoelastic wave equations,” Nonlin. Diffferent. Equat. Appl. (2011).

  26. G. Todorova, “Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms,” Compt. Rend. Acad. Sci. Sér. 1, 326, No. 2, 191–196 (1998).

    MathSciNet  MATH  Google Scholar 

  27. G. Todorova, “Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms,” J. Math. Anal. Appl., 239, No. 2, 213–226 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  28. E. Vitillaro, “Global nonexistence theorems for a class of evolution equations with dissipation,” Arch. Ration. Mech. Anal., 149, No. 2, 155–182 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  29. Z. Yang, “Blow-up of solutions for a class of nonlinear evolution equations with nonlinear damping and source terms,” Math. Meth. Appl. Sci., 25, No. 10, 825–833 (2002).

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Badji Mokhtar University, Annaba, Algeria

    D. Ouchenane

  2. Djillali Liabes University, Sidi Bel Abbes, Algeria

    Kh. Zennir

  3. University 20 Aout 55, Skikda, Algeria

    M. Bayoud

Authors
  1. D. Ouchenane
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kh. Zennir
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Bayoud
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 7, pp. 654–669, July, 2013.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ouchenane, D., Zennir, K. & Bayoud, M. Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms. Ukr Math J 65, 723–739 (2013). https://doi.org/10.1007/s11253-013-0809-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-013-0809-3

Keywords

Search

Navigation