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

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).