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This work is devoted to prove the exponential decay for the energy of solutions of the Korteweg-de Vries equation in a bounded interval with a localized damping term. Following the method in Menzala (2002) which combines energy estimates, multipliers and compactness arguments the problem is reduced to prove the unique continuation of weak solutions. In Menzala (2002) the case where solutions vanish on a neighborhood of both extremes of the bounded interval where equation holds was solved combining the smoothing results by T. Kato (1983) and earlier results on unique continuation of smooth solutions by J.C. Saut and B. Scheurer (1987). In this article we address the general case and prove the unique continuation property in two steps. We first prove, using multiplier techniques, that solutions vanishing on any subinterval are necessarily smooth. We then apply the existing results on unique continuation of smooth solutions.
@article{COCV_2005__11_3_473_0,
author = {Pazoto, Ademir Fernando},
title = {Unique continuation and decay for the {Korteweg-de} {Vries} equation with localized damping},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {473--486},
publisher = {EDP-Sciences},
volume = {11},
number = {3},
year = {2005},
doi = {10.1051/cocv:2005015},
mrnumber = {2148854},
zbl = {1148.35348},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2005015/}
}
TY - JOUR AU - Pazoto, Ademir Fernando TI - Unique continuation and decay for the Korteweg-de Vries equation with localized damping JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2005 SP - 473 EP - 486 VL - 11 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv:2005015/ DO - 10.1051/cocv:2005015 LA - en ID - COCV_2005__11_3_473_0 ER -
%0 Journal Article %A Pazoto, Ademir Fernando %T Unique continuation and decay for the Korteweg-de Vries equation with localized damping %J ESAIM: Control, Optimisation and Calculus of Variations %D 2005 %P 473-486 %V 11 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv:2005015/ %R 10.1051/cocv:2005015 %G en %F COCV_2005__11_3_473_0
Pazoto, Ademir Fernando. Unique continuation and decay for the Korteweg-de Vries equation with localized damping. ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 3, pp. 473-486. doi: 10.1051/cocv:2005015
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