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

Abstract

In this paper we are concerned with an 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 exponentially 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.