On the uniform decay for the Korteweg–de Vries equation with weak damping

Abstract

Abstract The aim of this work is to consider the Korteweg–de Vries equation in a finite interval with a very weak localized dissipation namely the H −1 ‐norm. Our main result says that the total energy decays locally uniform at an exponential rate. Our analysis improves earlier works on the subject ( Q. Appl. Math. 2002; LX (1):111–129; ESAIM Control Optim. Calculus Variations 2005; 11 (3):473–486) and gives a satisfactory answer to a problem suggested in ( Q. Appl. Math . 2002; LX (1):111–129). Copyright © 2007 John Wiley & Sons, Ltd.