It is established convergence to a particular equilibrium for weak solutions of abstract linear
equations of the second order in time associated with monotone operators with nontrivial kernel. Concerning nonlinear
hyperbolic equations with monotone and conservative potentials, it is proved a general asymptotic convergence result
in terms of weak and strong topologies of appropriate Hilbert spaces. It is also considered the stabilization of a
particular equilibrium via the introduction of an asymptotically vanishing restoring force into the evolution equation.
