On a model equation with small parameter as a coefficient of the second-order time derivative arising in the analysis of certain quasi-gasdynamics systems

We study a model nonlinear hyperbolic equation with small parameter as a coefficient of the second-order time derivative. We show that its long time dynamics are approximated in terms of global attractors by the dynamics of the limiting parabolic equation. 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.