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

TL;DR

This paper proves exponential decay of solutions to the KdV equation on the half-line with localized damping, in weighted Sobolev spaces, using Lyapunov methods, including decay of derivatives.

Contribution

It establishes uniform exponential stabilization in weighted Sobolev spaces for the KdV equation with localized damping on the half-line, extending previous results.

Findings

Solutions decay exponentially in weighted spaces

Decay of spatial derivatives of solutions

Applicable to damping active on 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 \cite{linares-pazoto} 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.