On the closeness of trajectories for model quasi-gasdynamic equations. Linear case

On a model example of a linear hyperbolic equation with small parameter multiplying the highest time derivative it is shown that the closeness of individual trajectories to the dynamics of the limiting parabolic equation essentially depends on the Fourier spectra of the initial data. We prove explicit estimates for the closeness of the solutions in the uniform norms and in the norms of the Sobolev spaces of arbitrary smoothness. We consider both the mutidimensional space-periodic case, and the case of a bounded domain.