Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping

Adriana Flores de Almeida; Marcelo Moreira Cavalcanti; Janaina Pedroso Zanchetta

doi:10.3934/eect.2019041

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2019, Volume 8, Issue 4: 847-865. Doi: 10.3934/eect.2019041

This issue Previous Article Optimal energy decay rates for some wave equations with double damping terms Next Article Existence and extinction in finite time for Stratonovich gradient noise porous media equations

Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping

Department of Mathematics, State University of Maringa, Maringa, 87020-900, Brazil

^* Corresponding author: janainazanchetta@yahoo.com.br
^* Corresponding author: janainazanchetta@yahoo.com.br

^* Research partially supported by the CNPq grant 300631/2003-0.
^** Research partially supported by CAPES

Received: October 2018

Revised: November 2018

Early access: June 2019

Published: June 2019

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Abstract

The following coupled damped Klein-Gordon-Schrödinger equations are considered

$ \begin{eqnarray*} i\psi_t + \Delta \psi + i \alpha b(x)(|\psi|^{2} + 1)\psi & = & \phi \psi \chi_{\omega} \; \hbox{in}\; \Omega \times (0, \infty), \; (\alpha >0)\ \\ \phi_{tt} - \Delta \phi + a(x) \phi_t & = & |\psi|^2 \chi_{\omega}\; \hbox{in}\; \Omega \times (0, \infty), \end{eqnarray*} $

where $ \Omega $ is a bounded domain of $ \mathbb{R}^2 $, with smooth boundary $ \Gamma $ and $ \omega $ is a neighbourhood of $ \partial \Omega $ satisfying the geometric control condition. Here $ \chi_{\omega} $ represents the characteristic function of $ \omega $. Assuming that $ a, b\in L^{\infty}(\Omega) $ are nonnegative functions such that $ a(x) \geq a_0 >0 $ in $ \omega $ and $ b(x) \geq b_{0} > 0 $ in $ \omega $, the exponential decay rate is proved for every regular solution of the above system. Our result generalizes substantially the previous ones given by Cavalcanti et. al in the reference [9] and [1].

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Mathematics Subject Classification: Primary: 35L70, 35B40.

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