Global Existence and Energy Decay of Solutions for a Nondissipative Wave Equation with a Time-Varying Delay Term

  • Abbes BenaissaEmail author
  • Salim A. Messaoudi
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 44)


We consider the energy decay for a nondissipative wave equation in a bounded domain with a time-varying delay term in the internal feedback. We use an approach introduced by Guesmia which leads to decay estimates (known in the dissipative case) when the integral inequalities method due to Haraux-Komornik (Haraux in Nonlinear Partial Differential Equations and Their Applications. Collège de France seminar, Vol. VII (Paris, 1983–1984), pp. 161–179, 1985; Komornik in Exact Controllability and Stabilization: The Multiplier Method, 1994) cannot be applied due to the lack of dissipativity. First, we study the stability of a nonlinear wave equation of the form in a bounded domain. We consider the general case with a nonlinear function h satisfying a smallness condition and obtain the decay of solutions under a relation between the weight of the delay term in the feedback and the weight of the term without delay. We impose no control on the sign of the derivative of the energy related to the above equation. In the second case we take θconst and h(∇u)=−∇Φ⋅∇u. We prove an exponential decay result of the energy without any smallness condition on Φ.


Nonlinear Wave Equation Decay Estimate Carleman Estimate Delay Term Observability Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We would like to thank very much the referees for their important remarks and comments which allow us to correct and improve this paper. This work has been partially funded by KFUPM under Project #FT111002.


  1. 1.
    Abdallah, C., Dorato, P., Benitez-Read, J., Byrne, R.: Delayed positive feedback can stabilize oscillatory system. In: American Control Conference, San Francisco, pp. 3106–3107 (1993) Google Scholar
  2. 2.
    Benaissa, A., Benazzouz, S.: Energy decay of solutions to the Cauchy problem for a nondissipative wave equation. J. Math. Phys. 51, 123504 (2010) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cavalcanti, M.M., Larkin, N.A., Soriano, J.A.: On solvability and stability of solutions of nonlinear degenerate hyperbolic equations with boundary damping. Funkc. Ekvacioj 41, 271–289 (1998) MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chen, G.: Control and stabilization for the wave equation in a bounded domain, part I. SIAM J. Control Optim. 17, 66–81 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Chen, G.: Control and stabilization for the wave equation in a bounded domain, part II. SIAM J. Control Optim. 19, 114–122 (1981) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    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. 24, 152–156 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Guesmia, A.: A new approach of stabilization of nondissipative distributed systems. SIAM J. Control Optim. 42, 24–52 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Guesmia, A.: Nouvelles inégalités intégrales et application à la stabilisation des systèmes distribués non dissipatifs. C. R. Math. Acad. Sci. Paris 336, 801–804 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Guesmia, A.: Inégalités intégrales et application à la stabilisation des systèmes distribués non dissipatifs. HDR thesis, Paul Verlaine-Metz University (2006) Google Scholar
  10. 10.
    Haraux, A.: Two remarks on dissipative hyperbolic problems. In: Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Vol. VII (Paris, 1983–1984). Research Notes in Mathematics, vol. 122, pp. 161–179. Pitman, Boston (1985) Google Scholar
  11. 11.
    Komornik, V.: Exact Controllability and Stabilization: The Multiplier Method. Masson, Paris (1994) zbMATHGoogle Scholar
  12. 12.
    Lasiecka, I., Triggiani, R.: Uniform exponential energy decay of wave equations in a bounded region with L 2(0,∞;L 2(Γ))-feedback control in the Dirichlet boundary conditions. J. Differ. Equ. 66, 340–390 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris (1969) zbMATHGoogle Scholar
  14. 14.
    Martinez, P.: A new method to obtain decay rate estimates for dissipative systems. ESAIM Control Optim. Calc. Var. 4, 419–444 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Nakao, M.: Decay of solutions of some nonlinear evolution equations. J. Math. Anal. Appl. 60, 542–549 (1977) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Nicaise, S., Pignotti, C.: Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 21, 935–958 (2008) MathSciNetzbMATHGoogle Scholar
  18. 18.
    Nicaise, S., Valein, J., Fridman, E.: Stability of the heat and of the wave equations with boundary time-varying delays. Discrete Contin. Dyn. Syst., Ser. S 2, 559–581 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Nicaise, S., Pignotti, C., Valein, J.: Exponential stability of the wave equation with boundary time-varying delay. Discrete Contin. Dyn. Syst., Ser. S 4, 693–722 (2011) MathSciNetzbMATHGoogle Scholar
  20. 20.
    Suh, I.H., Bien, Z.: Use of time delay action in the controller design. IEEE Trans. Autom. Control 25, 600–603 (1980) zbMATHCrossRefGoogle Scholar
  21. 21.
    Xu, C.Q., Yung, S.P., Li, L.K.: Stabilization of the wave system with input delay in the boundary control. ESAIM Control Optim. Calc. Var. 12, 770–785 (2006) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Laboratory of Analysis and Control of Partial Differential EquationsDjillali Liabes UniversitySidi Bel AbbesAlgeria
  2. 2.Department of Mathematics and StatisticsKFUPMDhahranSaudi Arabia

Personalised recommendations