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

The following coupled damped Klein-Gordon-Schrödinger equations are considered \begin{document}$ \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*} $\end{document} 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].