Well-posedness and stabilization of a model system for long waves posed on a quarter plane

TL;DR

This paper proves that a coupled KdV system modeling long wave propagation on a quarter plane is well-posed and exponentially stable when a localized damping term is applied, using Lyapunov and fixed point methods.

Contribution

It establishes the global well-posedness and exponential stability of a coupled KdV system with damping on a quarter plane, extending previous results to this specific setting.

Findings

Solutions are exponentially stable under damping.

The system is globally well-posed in weighted spaces.

Stability and well-posedness are proven using Lyapunov and fixed point methods.

Abstract

In this paper we are concerned with a initial boundary-value problem for a coupled system of two KdV equations, posed on the positive half line, under the effect of a localized damping term. The model arises when modeling the propagation of long waves generated by a wave maker in a channel. It is shown that the solutions of the system are exponential stable and globally well-posed in the weighted space $L^2(e^{2bx}dx)$ for $b>0$. The stabilization problem is studied using a Lyapunov approach while the well-posedness result is obtained combining fixed point arguments and energy type estimates.