Parametric exponential energy decay for dissipative electron-ion plasma waves

We consider the following evolution system of Klein-Gordon-Schrödinger type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{aligned} & i\psi _t + \kappa\psi _{xx} + i\alpha \psi = \phi \psi ,\,x \in \Omega ,\,t > 0, \\ & \phi _{tt} - \phi _{xx} + \phi + \lambda \phi _t = - Re \psi_{x}, \, x \in \Omega ,\,t > 0, \end{aligned} $$ \end{document} satisfying the following initial and boundary conditions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{aligned} & \psi (x,0) = \psi _0 (x),\,\phi (x,0) = \phi _0 (x),\,\phi _t (x,0) = \phi _1 (x),\,x \in \Omega , \\ & \psi (x,t) = \phi (x,t) = 0,\quad x \in \partial \Omega ,\,t > 0, \end{aligned} $$ \end{document} with κ, α, λ positive constants and ω a bounded subset of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}.$ \end{document} This system describes the nonlinear interaction between high frequency electron waves and low frequency ion plasma waves in a homogeneous magnetic field, adapted to model the UHH plasma heating scheme. The system focuses on the vital role of collisions, by considering the non-homogeneous polarization drift for the low frequency coupling. In Part I we set up the system, starting from first principles. In Part II we work out global existence and uniqueness of solutions and establish the necessary conditions for the system to manifest energy decay. In Part III the results are physically interpreted, providing a threshold of the effectiveness of UHH, in terms of the plasma variables.