Advertisement

Acta Mathematica Scientia

, Volume 39, Issue 1, pp 329–338 | Cite as

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

  • Shun-Tang Wu (吴舜堂)
Article
  • 3 Downloads

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.

Key words

blow up nonlinear source wave equation delay viscoelastic 

2010 MR Subject Classification

35B37 35L55 74D05 93D15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [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: 1043–1053MathSciNetCrossRefzbMATHGoogle Scholar
  2. [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: 697–713MathSciNetCrossRefzbMATHGoogle Scholar
  3. [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): 152–156MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Datko R. Two examples of ill-posedness with respect to time delays revisited. IEEE Trans Automatic Control, 1997, 42: 511–515MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Kafini M, Messaoudi S A. A blow-up result in a nonlinear wave equation with delay. Mediterr J Math, 2016, 13: 237–247MathSciNetCrossRefzbMATHGoogle Scholar
  6. [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: 13MathSciNetCrossRefzbMATHGoogle Scholar
  7. [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/s00033-011-0145-0Google Scholar
  8. [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: 105–117MathSciNetCrossRefzbMATHGoogle Scholar
  9. [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: 1890–1904MathSciNetCrossRefGoogle Scholar
  10. [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 284809Google Scholar
  11. [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): 2175–2190MathSciNetCrossRefzbMATHGoogle Scholar
  12. [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): 400–414MathSciNetCrossRefGoogle Scholar
  13. [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): 239–260MathSciNetCrossRefzbMATHGoogle Scholar
  14. [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): 299–323MathSciNetzbMATHGoogle Scholar
  15. [15]
    Messaoudi S A, Tatar N-e. Global existence and asymptotic behavior for a nonlinear viscoelastic problem. Mathematical Science Research Journal, 2003, 7(4): 136–149MathSciNetzbMATHGoogle Scholar
  16. [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): 509–526MathSciNetzbMATHGoogle Scholar
  17. [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): 1561–1585MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Nicaise S, Pignotti C. Stabilization of the wave equation with boundary or internal distributed delay. Diff Int Equs, 2008, 21(9/10): 935–958MathSciNetzbMATHGoogle Scholar
  19. [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): 425–479MathSciNetCrossRefzbMATHGoogle Scholar
  20. [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): 559–581MathSciNetCrossRefzbMATHGoogle Scholar
  21. [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): 693–722MathSciNetzbMATHGoogle Scholar
  22. [22]
    Pignotti C. A note on stabilization of locally damped wave equations with time delay. Syst Control Lett, 2012, 61(1): 92–97MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Song H T. Global nonexistence of positive initial energy solutions for a viscoelastic wave equation. Nonlinear Analysis: Theory, Methods & Applications, 2015, 125: 260–269MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Suh I H, Bien Z. Use of time delay action in the controller design. IEEE Trans Automat Control, 1980, 25: 600–603CrossRefzbMATHGoogle Scholar
  25. [26]
    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: 770–785MathSciNetzbMATHGoogle Scholar
  26. [26]
    Wu S T. Asymptotic behavior for a viscoelastic wave equation with a delay term. Taiwanese J Math, 2013, 17: 765–784MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Wuhan Institutes of Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  1. 1.General Education CenterNational Taipei University of TechnologyTaipeiChina

Personalised recommendations