  We consider a class of semi-linear dissipative hyperbolic equations in which
the operator associated to the linear part has a nontrivial kernel.
  Under appropriate assumptions on the nonlinear term, we prove that all
solutions decay to 0, as t -> +infinity, at least as fast as a suitable
negative power of t. Moreover, we prove that this decay rate is optimal in the
sense that there exists a nonempty open set of initial data for which the
corresponding solutions decay exactly as that negative power of t.
  Our results are stated and proved in an abstract Hilbert space setting, and
then applied to partial differential equations.
