Uniform Boundary Stabilization of the Wave Equation with a Nonlinear Delay Term in the Boundary Conditions

  • Wassila Ghecham
  • Salah-Eddine Rebiai
  • Fatima Zohra Sidiali
Conference paper
Part of the Trends in Mathematics book series (TM)


A wave equation in a bounded and smooth domain of \(\mathbb {R}^{n}\) with a delay term in the nonlinear boundary feedback is considered. Under suitable assumptions, global existence and uniform decay rates for the solutions are established by adopting an approach due to Lasiecka and Tataru (Differ Integral Equ 6:507–533, 1993). The proof of existence of solutions relies on a construction of a suitable approximating problem for which the existence of solution will be established using nonlinear semigroup theory and then passage to the limit gives the existence of solutions to the original problem. The uniform decay rates for the solutions are obtained by proving certain integral inequalities for the energy function and by establishing a comparison theorem which relates the asymptotic behaviour of the energy and of the solutions to an appropriate dissipative ordinary differential equation.


  1. 1.
    M.M. Cavalcanti, V.D. Cavalcanti, I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction. J. Differ. Equ. 236, 407–459 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    R. Datko, J. Lagnese, M. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 24, 152–156 (1986)MathSciNetCrossRefGoogle Scholar
  3. 3.
    V. Komornik, Exact Controllability and Stabilization. The Multiplier Method (Masson-John Wiley, Paris, 1994)Google Scholar
  4. 4.
    I. Lasiecka, D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differ. Integral Equ. 6, 507–533 (1993)MathSciNetzbMATHGoogle Scholar
  5. 5.
    I. Lasiecka, R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions. Appl. Math. Optim. 25, 189–244 (1992)MathSciNetCrossRefGoogle Scholar
  6. 6.
    S. Nicaise, C. Pignotti, 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)MathSciNetCrossRefGoogle Scholar
  7. 7.
    G.Q. Xu, S.P. Yung, L.K. Li, Stabilization of wave systems with input delay in the boundary control. ESAIM Control Optim. Calc. Var. 12, 770–785 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control Optim. 28, 466–477 (1990)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Wassila Ghecham
    • 1
  • Salah-Eddine Rebiai
    • 1
  • Fatima Zohra Sidiali
    • 1
  1. 1.LTMUniversity of Batna 2BatnaAlgeria

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