[Une remarque sur la stabilisation de l'équation des ondes 1-d]
Nous considérons l'équation des ondes sur un intervalle de longueur 1 avec un amortissement en un point ξ intérieur et avec la condition au bord de Dirichlet aux deux extrémités. Il est bien-connu que, si ξ est rationnel, l'énergie ne tend pas vers 0. Dans ce cas, nous prouvons que l'énergie décroît exponentiellement vers une constante que l'on explicitera.
We consider the wave equation on an interval of length 1 with an interior damping at ξ and with Dirichlet boundary condition at the two ends. It is well known that, if ξ is rational, the energy does not decay to 0. In this case, we prove that the energy decays exponentially to a constant which we identify.
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@article{CRMATH_2010__348_1-2_47_0,
author = {Nicaise, Serge and Valein, Julie},
title = {A remark on the stabilization of the 1-d wave equation},
journal = {Comptes Rendus. Math\'ematique},
pages = {47--51},
publisher = {Elsevier},
volume = {348},
number = {1-2},
year = {2010},
doi = {10.1016/j.crma.2009.11.015},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.11.015/}
}
TY - JOUR AU - Nicaise, Serge AU - Valein, Julie TI - A remark on the stabilization of the 1-d wave equation JO - Comptes Rendus. Mathématique PY - 2010 SP - 47 EP - 51 VL - 348 IS - 1-2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.11.015/ DO - 10.1016/j.crma.2009.11.015 LA - en ID - CRMATH_2010__348_1-2_47_0 ER -
%0 Journal Article %A Nicaise, Serge %A Valein, Julie %T A remark on the stabilization of the 1-d wave equation %J Comptes Rendus. Mathématique %D 2010 %P 47-51 %V 348 %N 1-2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.11.015/ %R 10.1016/j.crma.2009.11.015 %G en %F CRMATH_2010__348_1-2_47_0
Nicaise, Serge; Valein, Julie. A remark on the stabilization of the 1-d wave equation. Comptes Rendus. Mathématique, Tome 348 (2010) no. 1-2, pp. 47-51. doi : 10.1016/j.crma.2009.11.015. http://www.numdam.org/articles/10.1016/j.crma.2009.11.015/
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