  The general theory of Lyapunov's stability of first-order differential
inclusions in Hilbert spaces has been studied by the authors in a previous
work. This new contribution focuses on the natural case when the maximally
monotone operator governing the given inclusion has a domain with nonempty
interior. This setting permits to have nonincreasing Lyapunov functions on the
whole trajectory of the solution to the given differential inclusion. It also
allows some more explicit criteria for Lyapunov's pairs. Some consequences to
the viability of closed sets are given, as well as some useful cases relying on
the continuity or/and convexity of the involved functions. Our analysis makes
use of standard tools from convex and variational analysis.
