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First, we consider a semilinear hyperbolic equation with a locally distributed damping in a bounded domain. The damping is located on a neighborhood of a suitable portion of the boundary. Using a Carleman estimate [Duyckaerts, Zhang and Zuazua, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear); Fu, Yong and Zhang, SIAM J. Contr. Opt. 46 (2007) 1578-1614], we prove that the energy of this system decays exponentially to zero as the time variable goes to infinity. Second, relying on another Carleman estimate [Ruiz, J. Math. Pures Appl. 71 (1992) 455-467], we address the same type of problem in an exterior domain for a locally damped semilinear wave equation. For both problems, our method of proof is constructive, and much simpler than those found in the literature. In particular, we improve in some way on earlier results by Dafermos, Haraux, Nakao, Slemrod and Zuazua.
@article{COCV_2008__14_3_561_0,
author = {Tebou, Louis},
title = {A {Carleman} estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {561--574},
publisher = {EDP-Sciences},
volume = {14},
number = {3},
year = {2008},
doi = {10.1051/cocv:2007066},
mrnumber = {2434066},
zbl = {1146.93370},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2007066/}
}
TY - JOUR AU - Tebou, Louis TI - A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 561 EP - 574 VL - 14 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv:2007066/ DO - 10.1051/cocv:2007066 LA - en ID - COCV_2008__14_3_561_0 ER -
%0 Journal Article %A Tebou, Louis %T A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 561-574 %V 14 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv:2007066/ %R 10.1051/cocv:2007066 %G en %F COCV_2008__14_3_561_0
Tebou, Louis. A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 3, pp. 561-574. doi: 10.1051/cocv:2007066
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