Stabilization for the Wave Equation on Exterior Domains

  • L. Aloui
  • M. Khenissi
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 46)


The aim of this article is the study of stabilization for the wave equation on exterior domains, with Dirichlet boundary condition. More precisely, let O be a bounded, smooth domain of ℝ n (n odd); we consider the following wave equation on Ω = c Ō:
$$ \left( E \right)\left\{ \begin{gathered} \square u = \partial _t^2 u = 0 on \mathbb{R} \times \Omega \hfill \\ u\left( 0 \right) = f_1 ,\partial _t u\left( 0 \right) = f_2 on \Omega \hfill \\ u_{\partial \Omega \times \mathbb{R}} = 0 \hfill \\ \end{gathered} \right. $$
with the initial data f = (f 1, f 2) ∈ H(Ω) = H D × L 2, the completion of (C 0 (Ω))2 for the energy norm.


Wave Equation Exponential Decay Exterior Domain Finite Dimensional Space Carleman Estimate 
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.


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  1. [1]
    C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation, control and stabilisation of waves from the boundary, SIAM J. Control Optim. 305 (1992), 1024–1065.MathSciNetCrossRefGoogle Scholar
  2. [2]
    N. Burg, Decroissance de l’energie locale de l’equation des ondes pour le problemexterieur et absence de resonance au voisinage du reel, Acta Math. 180 (1998) 16–29.Google Scholar
  3. [3]
    P. Gerard, Microlocal defect measures, Comm. Partial Dif. Equations 16 (1991), 1761–1794.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    P. Gerard, Oscillations and concentrations effects in semi-linear dispersive wave equation, J. Fund. Analysis 41(1) (1996), 60–98.MathSciNetCrossRefGoogle Scholar
  5. [5]
    P.D. Lax, C.S. Morawetz, and R.S. Philips, Exponential decay of solution of the wave equation in the exterior of a star shaped obstacle, Comm. Pure. Appl. Math. 16 (1963) 477–486.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    RD. Lax and RS. Phillips, Scattering Theory Decay, Academic Press, New York, 1967.Google Scholar
  7. [7]
    G. Lebeau, Control for hyperbolic equations, Actes du Colloque de Saint Jean de Monts (1991).Google Scholar
  8. [8]
    G. Lebeau, Equations des ondes amorties, A. Boutet de Monvel and V. Marchenko (eds), Algebraic and Geometric Methods in Math. Physics,1996, Kluwer Academic Publisher, 73–109.Google Scholar
  9. [9]
    J.L. Lions, Controlabilite exacte, Perturbation et stabilisation des systemes distribues, R.M.A, Masson, 1988.Google Scholar
  10. [10]
    R. Melrose and J. Sjöstrand, Singularites of boundary value problems I, Comm. Pure. Appl. Math. 31 (1978), 593–617; II, C.P.A.M 35 (1982), 129–168.zbMATHGoogle Scholar
  11. [11]
    R. Melrose, Singularities and energy decay in acoustical scattering, Duke Math. J. 46 (1979) 43–59.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    C.S. Morawetz, Decay for solutions of the exterior problem for the wave equation, Comm. Pure. Appl. Math. 28 (1975), 229–264.MathSciNetzbMATHGoogle Scholar
  13. [13]
    C.S. Morawetz, J. Ralston, and W. Strauss, Decay of solutions of the wave equation outside non-trapping obstacles. Comm. Pure. Appl. Math. 30 (1977), 447–508.MathSciNetzbMATHGoogle Scholar
  14. [14]
    J. Ralston, Solution of the wave equation with localized energy, Comm. Pure. Appl. Math. 22 (1969), 807–823.MathSciNetzbMATHGoogle Scholar
  15. [15]
    J. Rauch and M. Taylor, Exponential decay of solutions for the hyperbolic equation in bouded domain, Indiana University Math. J. 24 (1972), 74–86.Google Scholar
  16. [16]
    W.A. Strauss, Dispersal of waves vanishing on the boundary of aexterior domain, Comm. Pure. Appl. Math. 28 (1975) 265–278.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • L. Aloui
    • 1
  • M. Khenissi
    • 1
  1. 1.Département de Mathématiques et InformatiqueFaculté des Sciences de GabèsGabèsTunisia

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