Remarks on weak stabilization of semilinear wave equations

Alain Haraux

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 6, page 553-560
  • ISSN: 1292-8119

Abstract

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If a second order semilinear conservative equation with esssentially oscillatory solutions such as the wave equation is perturbed by a possibly non monotone damping term which is effective in a non negligible sub-region for at least one sign of the velocity, all solutions of the perturbed system converge weakly to 0 as time tends to infinity. We present here a simple and natural method of proof of this kind of property, implying as a consequence some recent very general results of Judith Vancostenoble.

How to cite

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Haraux, Alain. "Remarks on weak stabilization of semilinear wave equations." ESAIM: Control, Optimisation and Calculus of Variations 6 (2010): 553-560. <http://eudml.org/doc/197309>.

@article{Haraux2010,
abstract = { If a second order semilinear conservative equation with esssentially oscillatory solutions such as the wave equation is perturbed by a possibly non monotone damping term which is effective in a non negligible sub-region for at least one sign of the velocity, all solutions of the perturbed system converge weakly to 0 as time tends to infinity. We present here a simple and natural method of proof of this kind of property, implying as a consequence some recent very general results of Judith Vancostenoble. },
author = {Haraux, Alain},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Weak stabilization; semilinear; wave equations.; semilinear conservative equation; essentially oscillatory solutions; nonmonotone damping term},
language = {eng},
month = {3},
pages = {553-560},
publisher = {EDP Sciences},
title = {Remarks on weak stabilization of semilinear wave equations},
url = {http://eudml.org/doc/197309},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Haraux, Alain
TI - Remarks on weak stabilization of semilinear wave equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 6
SP - 553
EP - 560
AB - If a second order semilinear conservative equation with esssentially oscillatory solutions such as the wave equation is perturbed by a possibly non monotone damping term which is effective in a non negligible sub-region for at least one sign of the velocity, all solutions of the perturbed system converge weakly to 0 as time tends to infinity. We present here a simple and natural method of proof of this kind of property, implying as a consequence some recent very general results of Judith Vancostenoble.
LA - eng
KW - Weak stabilization; semilinear; wave equations.; semilinear conservative equation; essentially oscillatory solutions; nonmonotone damping term
UR - http://eudml.org/doc/197309
ER -

References

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  1. L. Amerio and G. Prouse, Abstract almost periodic functions and functional equations. Van Nostrand, New-York (1971).  Zbl0215.15701
  2. J.M. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim.5 (1979) 169-179.  Zbl0405.93030
  3. M. Biroli, Sur les solutions bornées et presque périodiques des équations et inéquations d'évolution. Ann. Math. Pura Appl.93 (1972) 1-79.  Zbl0281.35006
  4. T. Cazenave and A. Haraux, Propriétés oscillatoires des solutions de certaines équations des ondes semi-linéaires. C. R. Acad. Sci. Paris Sér. I Math.298 (1984) 449-452.  Zbl0571.35074
  5. T. Cazenave and A. Haraux, Oscillatory phenomena associated to semilinear wave equations in one spatial dimension. Trans. Amer. Math. Soc.300 (1987) 207-233.  Zbl0628.35058
  6. T. Cazenave and A. Haraux, Some oscillatory properties of the wave equation in several space dimensions. J. Funct. Anal.76 (1988) 87-109.  Zbl0656.35099
  7. T. Cazenave, A. Haraux and F.B. Weissler, Une équation des ondes complètement intégrable avec non-linéarité homogène de degré 3. C. R. Acad. Sci. Paris Sér. I Math.313 (1991) 237-241.  
  8. T. Cazenave, A. Haraux and F.B. Weissler, A class of nonlinear completely integrable abstract wave equations. J. Dynam. Differential Equations5 (1993) 129-154.  Zbl0783.35003
  9. T. Cazenave, A. Haraux and F.B. Weissler, Detailed asymptotics for a convex hamiltonian system with two degrees of freedom. J. Dynam. Differential Equations5 (1993) 155-187.  Zbl0771.34038
  10. F. Conrad and M. Pierre, Stabilization of second order evolution equations by unbounded nonlinear feedbacks. Ann. Inst. H. Poincaré Anal. Non Linéaire11 (1994) 485-515.  Zbl0841.93028
  11. A. Haraux, Comportement à l'infini pour une équation des ondes non linéaire dissipative. C. R. Acad. Sci. Paris Sér. I Math.287 (1978) 507-509.  Zbl0396.35065
  12. A. Haraux, Comportement à l'infini pour certains systèmes dissipatifs non linéaires. Proc. Roy. Soc. Edinburgh Ser. A84 (1979) 213-234.  Zbl0429.35013
  13. A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations. J. Differential Equations59 (1985) 145-154.  Zbl0535.35006
  14. A. Haraux and V. Komornik, Oscillations of anharmonic Fourier series and the wave equation. Rev. Mat. Iberoamericana1 (1985) 57-77.  Zbl0612.35077
  15. A. Haraux, Semi-linear hyperbolic problems in bounded domains, Mathematical Reports Vol. 3, Part 1 , edited by J. Dieudonné. Harwood Academic Publishers, Gordon & Breach (1987).  Zbl0875.35054
  16. A. Haraux, Systèmes dynamiques dissipatifs et applications, R.M.A. 17, edited by Ph. Ciarlet and J.L. Lions. Masson, Paris (1990).  
  17. A. Haraux, Strong oscillatory behavior of solutions to some second order evolution equations, Publication du Laboratoire d'Analyse Numérique 94033, 10 p.  Zbl0816.35087
  18. B.M. Levitan and V.V. Zhikov, Almost periodic functions and differential equations. Cambridge University Press, Cambridge (1982).  Zbl0499.43005
  19. M. Slemrod, Weak asymptotic decay via a relaxed invariance principle for a wave equation with nonlinear, nonmonotone damping. Proc. Roy. Soc. Edinburgh Ser. A113 (1989) 87-97.  Zbl0699.35023
  20. J. Vancostenoble, Weak asymptotic stability of second order evolution equations by nonlinear and nonmonotone feedbacks. SIAM J. Math. Anal.30 (1998) 140-154.  Zbl0951.35092
  21. J. Vancostenoble, Weak asymptotic decay for a wave equation with weak nonmonotone damping, 17p (to appear).  Zbl0973.35131
  22. G.F. Webb, Compactness of trajectories of dynamical systems in infinite dimensional spaces. Proc. Roy. Soc. Edinburgh Ser. A84 (1979) 19-34.  Zbl0414.34042

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