We study the asymptotic behaviour as time goes
to infinity of bounded solutions to various types of non-local
in time problems. In particular we investigate
equations of order less than one, equations of order between one and two,
and second order equations with weak damping of memory type. We construct
appropriate Lyapunov functions and prove convergence to a steady state
using the Lojasiewicz technique. As an application we
also consider the non-isothermal Cahn-Hilliard equation with memory.