Stabilization of the Gear-Grimshaw system with weak damping

Abstract

The aim of this work is to consider the internal stabilization of a nonlinear coupled system of two Korteweg--de Vries equations in a finite interval under the effect of a very weak localized damping. The system was introduced by Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. Considering feedback controls laws and using Compactness--Uniqueness Argument, which reduce the problem to use a unique continuation property, we establish the exponential stability of the weak solutions when the exponent in the nonlinear term ranges over the interval $[1,4)$.

17 pages - Modifications on Abstract, in Remark 1.2, 1.3 and improvements on the Section 4

References in corpus (1)

• Control and Stabilization of the Korteweg-de Vries Equation on a Periodic Domain

Cited by in corpus (1)

• Weak damping for the Korteweg-de Vries equation