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 -> infinity this system turns into the coupling system of the parabolic and Schrodinger 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 -> infinity.