On a hyperbolic perturbation of a parabolic initial-boundary value problem

We deal with a linear parabolic initial-boundary value problem and its hyperbolic perturbation with a small parameter ε>0 in front of the 2nd order time derivative. We derive bounds for the perturbation error of the orders O(ε) and O(ε) in several norms in dependence with smoothness of data (without a priori conditions on solutions) but for non-smooth coefficients and keeping the free terms in equations as distributions. They essentially complement or improve some previously known bounds. In addition, we discuss regularizations of the initial time derivative.