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).
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The author thank the anonymous referee for their helpful comments and suggestions.
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This paper is dedicated to Maria Carolina.
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Capistrano–Filho, R.A. Stabilization of the Gear–Grimshaw System with Weak Damping. J Dyn Control Syst 24, 145–166 (2018). https://doi.org/10.1007/s10883-017-9363-x
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DOI: https://doi.org/10.1007/s10883-017-9363-x