On second-order parabolic and hyperbolic perturbations of a first-order hyperbolic system

We study the Cauchy problems for a first-order symmetric hyperbolic system of equations with variable coefficients and its singular perturbations that are second-order strongly parabolic and hyperbolic systems of equations with a small parameter $\tau>$ 0 in front of the second derivatives with respect to $x$ and $t$. The properties of solutions of all three systems are formulated, and estimates of order $O(\tau^{\alpha/2})$ are given for the difference between the solutions of the original system and systems with perturbations for an initial function $\mathbf{w}_0$ of smoothness $\alpha$ in the sense of $L^2(\mathbb{R}^n)$, 0 $<\alpha\le$ 2. For $\alpha$ = 1/2, a broad class of discontinuous functions $\mathbf{w}_0$ is covered. Applications to the linearized system of gas dynamics equations and to the linearized parabolic and hyperbolic second-order quasi-gasdynamic systems of equations are given.