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UNIFORM STABILIZATION IN WEIGHTED SOBOLEV SPACES FOR THE KDV EQUATION POSED ON THE HALFLINE
"... Abstract. Studied here is the largetime behavior of solutions of the Kortewegde Vries equation posed on the right halfline under the effect of a localized damping. Assuming as in [19] that the damping is active on a set (a0, +∞) with a0> 0, we establish the exponential decay of the solutions in ..."
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Abstract. Studied here is the largetime behavior of solutions of the Kortewegde Vries equation posed on the right halfline under the effect of a localized damping. Assuming as in [19] that the damping is active on a set (a0, +∞) with a0> 0, we establish the exponential decay of the solutions in the weighted spaces L 2 ((x + 1) m dx) for m ∈ 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. 1. Introduction. The Kortewegde Vries (KdV) equation was first derived as a model for the propagation of small amplitude long water waves along a channel [8, 14, 15]. It has been intensively studied from various aspects for both mathematics and physics since the 1960s when solitons were discovered through solving the KdV equation, and the inverse scattering method, a socalled nonlinear Fourier transform,