On a model system with a small parameter as a coefficient of the highest time derivative arising in the analysis of certain quasihydrodymamical systems

We study a model nonlinear hyperbolic system with a small parameter as a coefficient of the second-order time derivative. We show that its long time dynamics is approximated in terms of global attractors by the dynamics of the limiting parabolic system. The proximity of the individual trajectories essentially depends on their Fourier spectrum. The obtained results might be useful for the explanation of certain effects arising in the analysis of the quasi-gasdynamics systems.