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

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

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topPazoto, Ademir Fernando. "Unique continuation and decay for the Korteweg-de Vries equation with localized damping." ESAIM: Control, Optimisation and Calculus of Variations 11.3 (2010): 473-486. <http://eudml.org/doc/90773>.

@article{Pazoto2010,

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.; localized damping},

language = {eng},

month = {3},

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/90773},

volume = {11},

year = {2010},

}

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

DA - 2010/3//

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.; localized damping

UR - http://eudml.org/doc/90773

ER -

## References

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