Stabilization of the Korteweg-de Vries equation with localized damping

Abstract

We study the stabilization of solutions of the Korteweg-de Vries (KdV) equation in a bounded interval under the effect of a localized damping mechanism.Using multiplier techniques we deduce the exponential decay in time of the solutions of the underlying linear equation.A locally uniform stabilization result of the solutions of the nonlinear KdV model is also proved.The proof combines compactness arguments, the smoothing effect of the KdV equation on the line and unique continuation results. Introduction.We consider the Korteweg-de Vries (KdV) equation in a bounded interval (0, L) under the presence of a localized damping Ut + ux + uxxx + uux 4-a(x)u = 0 in (0, L) x (0, +oo), u(0,t) = u(L,t) = 0 for all t > 0, ux(L,t) = 0 for all t > 0, u(x, 0) = uq{x) in (0, L).Here a = a(x) is a nonnegative function belonging to L°°(0, L).In most of the paper we will also assume that a(x) > ao > 0 a.e. in an open, nonempty subset lo of (0,L).Therefore, the damping term is acting effectively in u.The KdV equation (1.1), in the absence of damping, models the (unidirectional) propagation of water waves of small amplitude in a bounded channel and has been the object of intensive research (see, for instance, [1], [13], and [24] and the references therein).