We review results on the existence and the long term behaviour of non--autonomous linear evolution equations. Emphasis is put on recent results on the asymptotic behaviour using a semigroup approach.

. We study retarded parabolic non--autonomous evolution equations whose coefficients converge as t ! 1 such that the autonomous problem in the limit has an exponential dichotomy. Then the non--autonomous problem inherits the exponential dichotomy and the solution of the inhomogeneous equation tends to the stationary solution at infinity. We use a generalized characteristic equation to deduce the exponential dichotomy and new representation formulas for the solution of the inhomogeneous equation. 1. Introduction In the present paper we continue the investigation of the long--term behaviour of asymptotically autonomous evolution equations begun in [30]. There we studied the Cauchy problem u(t) = A(t)u(t) + f(t); t ? s 0; u(s) = x; (1.1) on a Banach space X assuming that the linear operators A(t), t 0, are sectorial of the same type and satisfy the `Acquistapace--Terreni' condition (see (P) below) and that there exists another sectorial operator A such that lim t!1 R(w; A(t...

We show the (asymptotic) almost periodicity of the bounded solution to the parabolic evolution equation u (t) = A(t)u(t) +f(t) on R (on R+ ) assuming that the linear operators A(t) satisfy the `Acquistapace{Terreni' conditions, that the evolution family generated by A() has an exponential dichotomy, and that R(!; A()) and f are (asymptotically) almost periodic. 1.

The long term behaviour of autonomous linear Cauchy problems on a Banach space X has been studied systematically and quite successfully by means of spectral theory