  We consider a class of abstract second order evolution equations with a
restoring force that is strictly superlinear at infinity with respect to the
position, and a dissipation mechanism that is strictly superlinear at infinity
with respect to the velocity.
  Under the assumption that the growth of the restoring force dominates the
growth of the dissipation, we prove a universal bound property, namely that the
energy of solutions is bounded for positive times, independently of the initial
condition. Under a slightly stronger assumption, we show also a universal decay
property, namely that the energy decays (as time goes to infinity) at least as
a multiple of a negative power of $t$, again independent of the boundary
conditions.
  We apply the abstract results to solutions of some nonlinear wave, plate and
Kirchhoff equations in a bounded domain.
