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

Published online by Cambridge University Press:  15 July 2005

Ademir Fernando Pazoto
Ademir Fernando Pazoto*
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, PO Box 68530, CEP 21945-970, Rio de Janeiro, RJ, Brasil; ademir@acd.ufrj.br
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Abstract

This work is devoted to prove the exponential decay for the energyof solutions of the Korteweg-de Vries equation in a bounded intervalwith a localized damping term. Following the method in Menzala (2002)which combines energy estimates, multipliers and compactnessarguments the problem is reduced to prove the unique continuation ofweak solutions. In Menzala (2002) the case where solutions vanish on aneighborhood of both extremes of the bounded interval where equationholds was solved combining the smoothing results by T. Kato (1983)and earlier results on unique continuation of smooth solutions byJ.C. Saut and B. Scheurer (1987). In this article we address thegeneral case and prove the unique continuation property in twosteps. We first prove, using multiplier techniques, that solutionsvanishing on any subinterval are necessarily smooth. We then applythe existing results on unique continuation of smooth solutions.

Keywords

Unique continuationdecaystabilizationKdV equationlocalized damping.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 3 , July 2005 , pp. 473 - 486
Copyright
© EDP Sciences, SMAI, 2005

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