Singular limits of solutions to the Cauchy problem for second order linear differential equations in Hilbert spaces

We study the behavior of solutions to the problem
$$ \begin{cases} \varepsilon\Big(u''_\varepsilon(t)+A_1u_\varepsilon(t)\Big)+u'_\varepsilon(t)+ A_0u_\varepsilon(t)=f(t),\quad t>0,\\ u_\varepsilon(0)=u_0,\qquad u'_\varepsilon(0)=u_1, \end{cases} $$
in the Hilbert space $H$ as $\varepsilon\to0$, where $A_1$ and $A_0$ are two linear selfadjoint operators.