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

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 \cite{linares-pazoto} that the damping is active on a set $(a_0,+\mathrm{\infty })$ with $a_0>0$, we establish the exponential decay of the solutions in the weighted spaces $L^2((x+1{)}^{m}dx)$ for $m\in {\mathbb{N}}^{\ast }$ 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.