  We consider a second order equation with a linear "elastic" part and a
nonlinear damping term depending on a power of the norm of the velocity. We
investigate the asymptotic behavior of solutions, after rescaling them suitably
in order to take into account the decay rate and bound their energy away from
zero.We find a rather unexpected dichotomy phenomenon. Solutions with finitely
many Fouriercomponents are asymptotic to solutions of the linearized
equationwithout damping, and exhibit some sort of equipartition of theenergy
among the components. Solutions with infinitely manyFourier components tend to
zero weakly but not strongly. We showalso that the limit of the energy of
solutions depends only on thenumber of their Fourier components.The proof of
our results is inspired by the analysis of asimplified model which we devise
through an averaging procedure,and whose solutions exhibit the same asymptotic
properties as thesolutions to the original equation.
