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Decay Estimates for the Wave Equation with Internal Damping

  • Vilmos Komornik
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 118)

Abstract

We obtain optimal estimates for the solution of an integral inequality related to many stabilization problems. Then it is applied to improve some recent results concerning the energy decay of the wave equation with internal nonlinear feedback. Unlike the earlier works, our method also applies in the case of bounded feedback functions.

1991 Mathematics Subject Classification

35L05 93D15 

Key words and phrases

Wave equation feedback stabilization integral inequality 

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References

  1. 1.
    H. Brézis, Problèmesunilatéraux,J. Math. Pures Appl. 51(1972), 1–168.MathSciNetGoogle Scholar
  2. 2.
    F. Conrad, J. Leblond and J. P. Marmorat, Stabilizationof second order evolution equations by unbounded nonlinear feedback,toappear.Google Scholar
  3. 3.
    A. Haraux, Oscillations forcées pour certains systèmes dissipatifs non linéaires, preprint no 78010, Laboratoire d’AnalyseNumérique, Université Pierre et Marie Curie, Paris, 1978.Google Scholar
  4. 4.
    A. Haraux, Comportement à I’infini pour une équation d’ondes non linéairedissipative, C. R. Acad. Sci. ParisSér I Math. 287(1978), 507–509.MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. Haraux, “Semilinear hyperbolicproblems in bounded domains, Mathematical reports”, J. Dieudonné editor, Harwood Academic Publishers, Gordon andBreach, 1987.Google Scholar
  6. 6.
    A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Arch. Rat. Mech. Anal. (1988), 191–206.Google Scholar
  7. 7.
    V. Komornik, Estimation d’énergie pour quelques systèmesdissipatifs semilinéaires, C. R. Acad.Si. Paris Sér I Math. 314(1992), 747–751.MathSciNetzbMATHGoogle Scholar
  8. 8.
    V. Komornik, Stabilisation non linéaire de l’équation des ondeset d’un modèlede plaques, C. R. Acad. Sci.Paris Sér I Math. 315(1992),55–60.MathSciNetzbMATHGoogle Scholar
  9. 9.
    V. Komornik, On some integral and differentialinequalities,preprint no (1992) 490/P-283, IRMA, Université Louis Pasteur, Strasbourg, France.Google Scholar
  10. 10.
    J. Lagnese, Boundary stabilization of thin plates,SIAMStudies in Applied Mathematics 10, SIAM, Philadelphia, PA, 1989.Google Scholar
  11. 11.
    J.-L. Lions and W. A. Strauss, Some non-linearevolution equations,Bull. Soc. Math. Fance 93(1965), 43–96.MathSciNetzbMATHGoogle Scholar
  12. 12.
    P. Marcati, Decay and stability for nonlinearhyperbolic equations,J. Diff. Equations 55 (1984), 30–58.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    M. Nakao, On the decay of solutions of somenonlinear dissipative wave equations in higher dimensions,Math. Z. 193(1986),227–234.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    E. Zuazua, Stability and decay for a class ofnonlinear hyperbolic problems,Asymptotic Anal. 1, 2 (1988), 161–185.MathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 1994

Authors and Affiliations

  • Vilmos Komornik
    • 1
  1. 1.Institut de Recherche Mathématique AvancéeUniversité Louis Pasteur et C.N.R.S.Strasbourg CedexFrance

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