In a Hilbert space setting, we study the asymptotic behavior, as time $t$ goes to infinity, of the trajectories of a second-order differential equation governed by the Yosida regularization of a maximally monotone operator with time-varying positive index $\lambda(t)$. The dissipative and convergence properties are attached to the presence of a viscous damping term with positive coefficient $\gamma(t)$. A suitable tuning of the parameters $\gamma(t)$ and $\lambda(t)$ makes it possible to prove the weak convergence of the trajectories towards zeros of the operator. When the operator is the subdifferential of a closed convex proper function, we estimate the rate of convergence of the values. These results are in line with the recent articles by Attouch-Cabot, and Attouch-Peypouquet. In this last paper, the authors considered the case $\gamma (t) =\frac{\alpha}{t}$, which is naturally linked to Nesterov's accelerated method. We unify, and often improve the results already present in the literature.