Skip to main content
SpringerLink
Log in

Blow-Up of Solution for A Viscoelastic Wave Equation with Delay

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

In this paper, we consider the following viscoelastic wave equation with delay \({\left| {{u_t}} \right|^\rho }{u_{tt}} - \Delta u - \Delta {u_{tt}} + \int_0^t {g\left( {t - s} \right)} \Delta u\left( s \right)ds + {\mu _1}{\mu _t}\left( {x,t} \right) + {\mu _2}{\mu _t}\left( {x,t - \tau } \right) = b{\left| u \right|^{p - 2}}u\) in a bounded domain. Under appropriate conditions on μ1, μ2, the kernel function g, the nonlinear source and the initial data, there are solutions that blow up in finite time.

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. Cavalcanti M M, Domingos Cavalcanti V N, Ferreira J. Existence and uniform decay of nonlinear viscoelastic equation with strong damping. Math Methods Appl Sci, 2001, 24: 1043–1053

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Datko R, Lagnese J, Polis M P. An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J Control Optim, 1986, 24(1): 152–156

    Article  MathSciNet  MATH  Google Scholar 

  4. Datko R. Two examples of ill-posedness with respect to time delays revisited. IEEE Trans Automatic Control, 1997, 42: 511–515

    Article  MathSciNet  MATH  Google Scholar 

  5. Kafini M, Messaoudi S A. A blow-up result in a nonlinear wave equation with delay. Mediterr J Math, 2016, 13: 237–247

    Article  MathSciNet  MATH  Google Scholar 

  6. Kafini M, Messaoudi S A, Nicaise S. A blow-up result in a nonlinear abstract evolution system with delay. NoDEA Nonlinear Differential Equations Appl, 2016, 23: 13

    Article  MathSciNet  MATH  Google Scholar 

  7. Kirane M, Belkacem S H. Existence and asymptotic stability of a viscoelastic wave equation with a delay. Z Angew Math Phys, DOI 10.1007/s00033-011-0145-0

  8. Li J, Chai S G. Energy decay for a nonlinear wave equation of variable coefficients with acoustic boundary conditions and a time-varying delay in the boundary feedback. Nonlinear Analysis: Theory, Methods & Applications, 2015, 112: 105–117

    Article  MathSciNet  MATH  Google Scholar 

  9. LiuW J. General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source. Nonlinear Analysis: Theory, Methods & Applications, 2010, 73: 1890–1904

    Article  MathSciNet  Google Scholar 

  10. Liu W J, Li G, Hong L H. General decay and blow-up of solutions for a system of viscoelastic equations of Kirchhoff type with strong damping. Journal of Function Spaces, 2014, 2014: Article ID 284809

  11. Liu W J, Yu J. On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms. Nonlinear Analysis: Theory, Methods & Applications, 2011, 74(6): 2175–2190

    Article  MathSciNet  MATH  Google Scholar 

  12. Liu W J, Chen K, Yu J. Asymptotic stability for a non-autonomous full von Karman beam with thermoviscoelastic damping. Applicable Analysis, 2018, 97(3): 400–414

    Article  MathSciNet  Google Scholar 

  13. Liu W J, Zhu B Q, Li G, Wang D H. General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term. Evolution Equations and Control Theory, 2017, 6(2): 239–260

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu W J, Sun Y, Li G. On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with a nonlinear source term. Topological Methods in Nonlinear Analysis, 2017, 49(1): 299–323

    MathSciNet  MATH  Google Scholar 

  15. Messaoudi S A, Tatar N-e. Global existence and asymptotic behavior for a nonlinear viscoelastic problem. Mathematical Science Research Journal, 2003, 7(4): 136–149

    MathSciNet  MATH  Google Scholar 

  16. Mohamed Ferhat, Ali Hakem. Asymptotic behavior for a weak viscoelastic wave equations with a dynamic boundary and time varying delay term. J Appl Math Comput, 2016, 51(1/2): 509–526

    MathSciNet  MATH  Google Scholar 

  17. 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, 2006, 45(5): 1561–1585

    Article  MathSciNet  MATH  Google Scholar 

  18. Nicaise S, Pignotti C. Stabilization of the wave equation with boundary or internal distributed delay. Diff Int Equs, 2008, 21(9/10): 935–958

    MathSciNet  MATH  Google Scholar 

  19. Nicaise S, Valein J. Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks. Netw Heterog Media, 2007, 2(3): 425–479

    Article  MathSciNet  MATH  Google Scholar 

  20. Nicaise S, Valein J, Fridman E. Stabilization of the heat and the wave equations with boundary time-varying delays. Discrete and Continuous Dynamical Systems - Series S, 2009, 2(3): 559–581

    Article  MathSciNet  MATH  Google Scholar 

  21. Nicaise S, Pignotti C, Valein J. Exponential stability of the wave equation with boundary time-varying delay. Discrete and Continuous Dynamical Systems - Series S, 2011, 4(3): 693–722

    MathSciNet  MATH  Google Scholar 

  22. Pignotti C. A note on stabilization of locally damped wave equations with time delay. Syst Control Lett, 2012, 61(1): 92–97

    Article  MathSciNet  MATH  Google Scholar 

  23. Song H T. Global nonexistence of positive initial energy solutions for a viscoelastic wave equation. Nonlinear Analysis: Theory, Methods & Applications, 2015, 125: 260–269

    Article  MathSciNet  MATH  Google Scholar 

  24. Suh I H, Bien Z. Use of time delay action in the controller design. IEEE Trans Automat Control, 1980, 25: 600–603

    Article  MATH  Google Scholar 

  25. Xu G Q, Yung S P, Li L K. Stabilization of the wave systems with input delay in the boundary control. ESAIM: Control Optim Calc Var, 2006, 12: 770–785

    MathSciNet  MATH  Google Scholar 

  26. Wu S T. Asymptotic behavior for a viscoelastic wave equation with a delay term. Taiwanese J Math, 2013, 17: 765–784

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. General Education Center, National Taipei University of Technology, Taipei, Taiwan, 106, China

    Shun-Tang Wu  (吴舜堂)

Authors
  1. Shun-Tang Wu  (吴舜堂)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shun-Tang Wu  (吴舜堂).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, ST. Blow-Up of Solution for A Viscoelastic Wave Equation with Delay. Acta Math Sci 39, 329–338 (2019). https://doi.org/10.1007/s10473-019-0124-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-019-0124-7

Key words

2010 MR Subject Classification

Search

Navigation