Uniform stabilization of a class of coupled systems of KdV equations with localized damping

Abstract

Abstract. We study the stabilization of solutions of a coupled system of Kortewegde Vries (KdV) equations in a bounded interval under the effect of a localized damping term. We use multiplier techniques combined with the so-called "compactness-uniqueness argument". The problem is then reduced to proving a unique continuation property (UCP) for weak solutions. The exponential decay of solutions was previously obtained in Bisognin, Bisognin, and Menzala (2003) when the damping was effective simultaneously in neighborhoods of both extremes of the bounded interval. In this work we address the general case using a different approach to obtain the UCP and stabilize the system.

Uniform stabilization of a nonlinear dispersive system

“…It is also worth mentioning some results obtained for the scalar KdV equation in connection with the analysis developed here [10,12,13,16,22,[24][25][26]. In fact, in system (1.2), the problem was addressed only in a bounded domain [3,15,21], and estimate (1.6) was proved following the ideas introduced for the analysis of the corresponding scalar case. The proof of our main result combines multipliers and the so-called compactnessuniqueness argument that leads one to apply a unique continuation result.…”

Well-posedness and stabilization of a model system for long waves posed on a quarter plane

“…We finish this section mentioning some results obtained for the scalar KdV equation in connection with the analysis developed here [6,11,13,14,17,23,25,26,27,28]. In what concern system (1.1), except in [22], the problem on internal stabilization was addressed only on a bounded domain [3,16,21]. In fact, most of results present in these works are proved following the ideas introduced for the analysis of the corresponding scalar case.…”