Uniform stabilization in weighted Sobolev spacesfor the KdV equation posed on the half-line

Ademir Fernando Pazoto; Lionel Rosier

doi:10.3934/dcdsb.2010.14.1511

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2010, Volume 14, Issue 4: 1511-1535. Doi: 10.3934/dcdsb.2010.14.1511

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Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line

Ademir Fernando Pazoto^1, and
Lionel Rosier^2,

1.
Instituto de Matemática, Universidade Federal do Rio de Janeiro, P.O. Box 68530, CEP 21941-909, Rio de Janeiro, RJ
2.
Institut Élie Cartan, UMR 7502 UHP/CNRS/INRIA, B.P. 239, F-54506 Vandoeuvre-lès-Nancy Cedex

Received: July 31, 2009

Revised: November 30, 2009

Published: May 2010

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Abstract

Studied here is the large-time behavior of solutions of the Korteweg-de Vries equation posed on the right half-line under the effect of a localized damping. Assuming as in [19] that the damping is active on a set $(a_0,+\infty)$ with $a_0>0$, we establish the exponential decay of the solutions in the weighted spaces $L^2((x+1)^mdx)$ for $m\in $N^* and $L^2(e^{2bx}dx)$ for $b>0$ by a Lyapunov approach. The decay of the spatial derivatives of the solution is also derived.

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Mathematics Subject Classification: Primary: 93D15, 35Q53; Secondary: 93C20.

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Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line

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