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.

On the controllability and stabilization of the linearized Benjamin-Ono equation

“…As in the case of the controllability, the stabilization has been widely studied in the last few years. For instance, for the KdV equation we refer to the works of Russell and Zhang [23] and [24] for the periodic case with a feedback law and the work of Perla, Vasconcellos and Zuazua [20] for the case of a linear localized damping in a bounded domain.…”

Stabilization of the Kawahara equation with localized damping

“…In [25] this result was proved by excluding a countable set of critical lengths L but later on in [26] it was shown that the same holds for all L, unlike the case of the KdV in which there is effectively a countable set of lengths for which the energy of the corresponding linear system does not decay (see [15,19]). …”

“…Nonlinear dispersive problems have been object of intensive research (see, for instance, the classical paper of Benjamin et al [1], Biagioni and Linares [3], Bona and Chen [4], Menzala et al [15], Rosier [19], and references therein). Recently global stabilization of the generalized KdV system has been obtained by Rosier and Zhang [20].…”

“…In this paper, following from close the works by Menzala et al [15] and Rosier [19] on the KdV equation, we study the problem of the decay of the energy for the (K) system as t → +∞, in the presence of the localized damping term a(x)u.…”

The Linear Kawahara Equation on the Half-Line

“…In the case with periodic boundary conditions, see [14]. Works involving the KdV system in bounded domain can be cited, such as [7] and [8,11,12]. In unbounded domain we have, for instance, the following articles: [6,9].…”