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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  22. G.F. Webb, Compactness of trajectories of dynamical systems in infinite dimensional spaces. Proc. Roy. Soc. Edinburgh Ser. A84 (1979) 19-34.  

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