Unique continuation and decay for the Korteweg-de Vries equation with localized damping

Instituto de Matemática, Universidade Federal do Rio de Janeiro, PO Box 68530, CEP 21945-970, Rio de Janeiro, RJ, Brasil; ademir@acd.ufrj.br

Received: 14 October 2003
Revised: 19 October 2004

Abstract

This work is devoted to prove the exponential decay for the energy of solutions of the Korteweg-de Vries equation in a bounded interval with a localized damping term. Following the method in Menzala (2002) which combines energy estimates, multipliers and compactness arguments the problem is reduced to prove the unique continuation of weak solutions. In Menzala (2002) the case where solutions vanish on a neighborhood of both extremes of the bounded interval where equation holds was solved combining the smoothing results by T. Kato (1983) and earlier results on unique continuation of smooth solutions by J.C. Saut and B. Scheurer (1987). In this article we address the general case and prove the unique continuation property in two steps. We first prove, using multiplier techniques, that solutions vanishing on any subinterval are necessarily smooth. We then apply the existing results on unique continuation of smooth solutions.