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

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), +infinity) with a(0) > 0, we establish the exponential decay of the solutions in the weighted spaces L-2((x + 1)(m)dx) for m is an element of 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.