Unique continuation and decay for the Korteweg-de Vries equation with localized damping

Ademir Fernando Pazoto

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • Volume: 11, Issue: 3, page 473-486
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Pazoto, Ademir Fernando. "Unique continuation and decay for the Korteweg-de Vries equation with localized damping." ESAIM: Control, Optimisation and Calculus of Variations 11.3 (2005): 473-486. <http://eudml.org/doc/244711>.

@article{Pazoto2005,
abstract = {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.},
author = {Pazoto, Ademir Fernando},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {unique continuation; decay; stabilization; KdV equation; localized damping; Unique continuation},
language = {eng},
number = {3},
pages = {473-486},
publisher = {EDP-Sciences},
title = {Unique continuation and decay for the Korteweg-de Vries equation with localized damping},
url = {http://eudml.org/doc/244711},
volume = {11},
year = {2005},
}

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
PB - EDP-Sciences
VL - 11
IS - 3
SP - 473
EP - 486
AB - 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.
LA - eng
KW - unique continuation; decay; stabilization; KdV equation; localized damping; Unique continuation
UR - http://eudml.org/doc/244711
ER -

References

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