  We consider reaction-diffusion systems and other related dissipative systems
on unbounded domains which would have a Liapunov function (and gradient
structure) when posed on a finite domain. In this situation, the system may
reach local equilibrium on a rather fast time scale but the infinite amount of
mass or energy leads to persistent mass or energy flow for all times. In
suitably rescaled variables the system converges to a steady state that
corresponds to asymptotically self-similar behavior in the original system.
