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Stabilization of the Kawahara equation with localized damping

Published online by Cambridge University Press:  30 October 2009

Carlos F. Vasconcellos  and
Patricia N. da Silva
Carlos F. Vasconcellos
Affiliation:
Instituto de Matemática e Estatística – UERJ, 524 R. São Francisco Xavier, Sala 6016, Bloco D – CEP 20550-013, Rio de Janeiro, Brazil. cfredvasc@ime.uerj.br; nunes@ime.uerj.br
Patricia N. da Silva
Affiliation:
Instituto de Matemática e Estatística – UERJ, 524 R. São Francisco Xavier, Sala 6016, Bloco D – CEP 20550-013, Rio de Janeiro, Brazil. cfredvasc@ime.uerj.br; nunes@ime.uerj.br
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Abstract

We study the stabilization of global solutions of the Kawahara (K) equation in a bounded interval, under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. The proof requires of a unique continuation theorem and the smoothing effect of the (K) equation on the real line, which are proved in this work.

Keywords

Kawahara equationstabilizationenergy decaylocalized damping
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 1 , January 2011 , pp. 102 - 116
Copyright
© EDP Sciences, SMAI, 2009

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