Uniform stabilization of the Klein-Gordon system

Marcelo M. Cavalcanti; Leonel G. Delatorre; Daiane C. Soares; Victor Hugo Gonzalez Martinez; Janaina P. Zanchetta

doi:10.3934/cpaa.2020230

Article Contents

2020, Volume 19, Issue 11: 5131-5156. Doi: 10.3934/cpaa.2020230

This issue Previous Article Normalized solutions for 3-coupled nonlinear Schrödinger equations Next Article Quasi-periodic solutions for the two-dimensional systems with an elliptic-type degenerate equilibrium point under small perturbations

Uniform stabilization of the Klein-Gordon system

1.
Department of Mathematics, State University of Maringá, 87020-900, Maringá, Brazil
2.
Department of Mathematics, Federal University of Pampa - Campus Itaqui, 97650-000, Itaqui, Brazil
3.
Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil

^* Corresponding author
^* Corresponding author

Received: October 2019

Revised: June 2020

Early access: September 2020

Published: November 2020

Research of Marcelo M. Cavalcanti is partially supported by the CNPq Grant 300631/2003-0. Research of Victor Hugo Gonzalez Martinez is partially supported by CAPES

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Abstract

We consider the Klein-Gordon system posed in an inhomogeneous medium $ \Omega $ with smooth boundary $ \partial \Omega $ subject to two localized dampings. The first one is of the type viscoelastic and is distributed around a neighborhood $ \omega $ of the boundary according to the Geometric Control Condition. The second one is a frictional damping and we consider it hurting the geometric condition of control. We show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space. For this purpose, refined microlocal analysis arguments are considered by exploiting ideas due to Burq and Gérard [5]. Although the present problem has some similarity to the reference [6] it is important to mention that due to the Kelvin-Voigt dissipation character associated with the nonlinearity of the problem the approach used is completely new, which is the main purpose of this paper.

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Mathematics Subject Classification: Primary: 35L05; Secundary: 35L53, 35B40, 93B07.

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Figure 1. The Kelvin-Voigt dampings act in $O_1 = \Omega \backslash A$ and $O_2 = \Omega \backslash B$ while the frictional dampings are effective in a collar of $\partial A$ and $\partial B$

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Figure 2. Admissible geometries for the Kelvin-Voigt and frictional dissipations $ a(x) $ and $ \gamma_1(x) $, respectively

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Figure 3. Admissible geometries for the Kelvin-Voigt and frictional dissipations $ a(x) $ and $ \gamma_1(x) $, respectively

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