Limits of solutions to the singularly perturbed abstract hyperbolic-parabolic system

We study the behavior of solutions to the problem
$$ \left\{
\begin{array}{l} \varepsilon u''_\varepsilon(t)+u'_\varepsilon(t)+A(t)u _\varepsilon(t)=f_\varepsilon(t),\quad t\in(0,T),\\ u_\varepsilon(0)=u_{0\varepsilon},\quad u'_\varepsilon(0)=u_{1\varepsilon}, \end{array}
\right. $$
in the Hilbert space $\mathrm H$ as $\varepsilon\to0$, where $A(t)$, $t\in(0,\infty)$, is a family of linear self-adjoint operators.