  We consider an abstract second order evolution equation with damping. The
"elastic" term is represented by a self-adjoint nonnegative operator A with
discrete spectrum, and the nonlinear term has order greater than one at the
origin. We investigate the asymptotic behavior of solutions.
  We prove the coexistence of slow solutions and fast solutions. Slow solutions
live close to the kernel of A, and decay as negative powers of t as solutions
of the first order equation obtained by neglecting the operator A and the
second order time-derivatives in the original equation. Fast solutions live
close to the range of A and decay exponentially as solutions of the linear
homogeneous equation obtained by neglecting the nonlinear terms in the original
equation.
  The abstract results apply to semilinear dissipative hyperbolic equations.
