On the Well-posedness and Asymptotic Behavior of a Nonlinear Dispersive System in Weighted Spaces

Abstract

This paper is concerned with a model for propagation of long waves in a channel generated by a wave maker mounted at one end. The mathematical structure consists in a coupled system of two nonlinear Korteweg-de Vries equations posed on the positive half line. Under the effect of a localized damping term it is shown that the solutions of the system are exponentially stable and globally well-posed in the weighted space L2((x+1)mdx) for m≥1. The stabilization problem is studied constructing a Lyapunov function by induction on m and the well-posedness is obtained by passing to the limit in a sequence of solutions in L2(e2bxdx) for b>0.