Linear singular perturbations of hyperbolic-parabolic type

We study the behavior of solutions of the problem $\varepsilon u''(t)+u'(t)+Au(t)=f(t)$, $u(0)=u_0$, $u'(0)=u_1$ in the Hilbert space $H$ as $\varepsilon\to 0$, where $A$ is a linear, symmetric, strong positive operator.