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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  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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