Remarks on weak stabilization of semilinear wave equations

Alain Haraux

ESAIM: Control, Optimisation and Calculus of Variations (2001)

  • 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 (2001): 553-560. <http://eudml.org/doc/90608>.

@article{Haraux2001,
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},
pages = {553-560},
publisher = {EDP-Sciences},
title = {Remarks on weak stabilization of semilinear wave equations},
url = {http://eudml.org/doc/90608},
volume = {6},
year = {2001},
}

TY - JOUR
AU - Haraux, Alain
TI - Remarks on weak stabilization of semilinear wave equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2001
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/90608
ER -

References

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  1. [1] L. Amerio and G. Prouse, Abstract almost periodic functions and functional equations. Van Nostrand, New-York (1971). Zbl0215.15701MR275061
  2. [2] J.M. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim. 5 (1979) 169-179. Zbl0405.93030MR533618
  3. [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. [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. Zbl0571.35074MR750743
  5. [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. Zbl0628.35058MR871673
  6. [6] T. Cazenave and A. Haraux, Some oscillatory properties of the wave equation in several space dimensions. J. Funct. Anal. 76 (1988) 87-109. Zbl0656.35099MR923046
  7. [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. Zbl0755.35009MR1126387
  8. [8] T. Cazenave, A. Haraux and F.B. Weissler, A class of nonlinear completely integrable abstract wave equations. J. Dynam. Differential Equations 5 (1993) 129-154. Zbl0783.35003MR1205457
  9. [9] T. Cazenave, A. Haraux and F.B. Weissler, Detailed asymptotics for a convex hamiltonian system with two degrees of freedom. J. Dynam. Differential Equations 5 (1993) 155-187. Zbl0771.34038MR1205458
  10. [10] F. Conrad and M. Pierre, Stabilization of second order evolution equations by unbounded nonlinear feedbacks. Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994) 485-515. Zbl0841.93028MR1302277
  11. [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. [12] A. Haraux, Comportement à l’infini pour certains systèmes dissipatifs non linéaires. Proc. Roy. Soc. Edinburgh Ser. A 84 (1979) 213-234. Zbl0429.35013
  13. [13] A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations. J. Differential Equations 59 (1985) 145-154. Zbl0535.35006MR804885
  14. [14] A. Haraux and V. Komornik, Oscillations of anharmonic Fourier series and the wave equation. Rev. Mat. Iberoamericana 1 (1985) 57-77. Zbl0612.35077MR850409
  15. [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). Zbl0875.35054MR1078761
  16. [16] A. Haraux, Systèmes dynamiques dissipatifs et applications, R.M.A. 17, edited by Ph. Ciarlet and J.L. Lions. Masson, Paris (1990). Zbl0726.58001MR1084372
  17. [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. [18] B.M. Levitan and V.V. Zhikov, Almost periodic functions and differential equations. Cambridge University Press, Cambridge (1982). Zbl0499.43005MR690064
  19. [19] M. Slemrod, Weak asymptotic decay via a relaxed invariance principle for a wave equation with nonlinear, nonmonotone damping. Proc. Roy. Soc. Edinburgh Ser. A 113 (1989) 87-97. Zbl0699.35023MR1025456
  20. [20] J. Vancostenoble, Weak asymptotic stability of second order evolution equations by nonlinear and nonmonotone feedbacks. SIAM J. Math. Anal. 30 (1998) 140-154. Zbl0951.35092MR1646669
  21. [21] J. Vancostenoble, Weak asymptotic decay for a wave equation with weak nonmonotone damping, 17p (to appear). Zbl0973.35131
  22. [22] G.F. Webb, Compactness of trajectories of dynamical systems in infinite dimensional spaces. Proc. Roy. Soc. Edinburgh Ser. A 84 (1979) 19-34. Zbl0414.34042MR549869

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