The limit behaviour of solutions of a singularly perturbed system is examined in the case where the fast flow need not converge to a stationary point. The topological convergence as well as information about the distribution of the values of the solutions can be determined in the case that the support of the limit invariant measure of the fast flow is an asymptotically stable attractor.
We consider second order quasilinear evolution equations where also the main part contains functional dependence on the unknown function. First, existence of solutions in (0, T ) is proved and examples satisfying the assumptions of the existence theorem are formulated. Then a uniqueness theorem is proved. Finally, existence and some qualitative properties of the solutions in (0,∞) (boundedness and stabilization as t → ∞) are shown.