Exponential Decay Rates for the Damped Korteweg-de Vries Type Equation

TL;DR

This paper establishes the exponential decay rate of the L^2-norm for a damped Korteweg-de Vries equation on the real line and shows that the H^1-norm does not necessarily decay for arbitrary initial data.

Contribution

It provides the first proof of exponential decay for the L^2-norm in the damped KdV equation and clarifies the limitations of decay in the H^1-norm.

Findings

L^2-norm decays exponentially over time

H^1-norm decay is not guaranteed for all initial data

Decay rates depend on the damping and initial conditions

Abstract

The exponential decay rate of $L^2-$norm related to the Korteweg-de Vries equation with localized damping posed on whole real line will be established. In addition, by using classical arguments we determine the $H^1-$norm of the solution associated to Korteweg-de Vries equation with damping in whole domain, can not have a decay property for an arbitrary initial data.