November  2010, 14(4): 1511-1535. doi: 10.3934/dcdsb.2010.14.1511

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

1. 

Instituto de Matemática, Universidade Federal do Rio de Janeiro, P.O. Box 68530, CEP 21941-909, Rio de Janeiro, RJ, Brazil

2. 

Institut Élie Cartan, UMR 7502 UHP/CNRS/INRIA, B.P. 239, F-54506 Vandoeuvre-lès-Nancy Cedex

Received  August 2009 Revised  December 2009 Published  August 2010

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.
Citation: Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511
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