A basic question is to establish convergence to equilibrium for globally defined solutions to evolution problems. The purpose here is to emphasize the role of symmetry. In particular, it is proved that in some cases the {em first moments of energy} are constant on the $omega$-limit set of the solution. This key property is used to prove convergence in two model evolution problems. This communication is based on two joint works with P. Felmer cite{busca_felmer} and M.A. Jendoubi, P. Polacik cite{busca_jendoubi}.