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Stabilization of the Gear-Grimshaw system with weak damping

By R. A. Capistrano-Filho


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 $[1,4)$.Comment: 17 pages - Modifications on Abstract, in Remark 1.2, 1.3 and improvements on the Section

Topics: Mathematics - Analysis of PDEs, 35Q53 (Primary), 37K10, 93B05, 93D15 (Secondary)
Year: 2017
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