The Singular Limit of the Dissipative Zakharov System

The dissipative Zakharov system which models the propagation of Langmuir waves in plasmas is considered on the interval $[0,L]$. We are interested in the case of large ion acoustic speed $\lambda$. After the formal limiting transition $\lambda\to\infty$ this system turns into the coupling system of the parabolic and Schrödinger equations. We prove that this limit system has a solution and generates a dissipative dynamical system possessing a global compact attractor. Our main result is the upper semicontinuity of the attractor as $\lambda\to\infty$.