Asymptotically autonomous parabolic evolution equations

Abstract.

We study the long-term behaviour of the parabolic evolution equation $\[u'(t)=A(t)u(t)+f(t), t>s,\quad u(s)=x. \]$ If $A(t)$ converges to a sectorial operator A with $\sigma(A)\cap i \Bbb R =\emptyset$ as $t\to\infty$, then the evolution family solving the homogeneous problem has exponential dichotomy. If also $f(t)\to f_\infty$, then the solution u converges to the 'stationary solution at infinity', i.e., $\lim_{t\to\infty}u(t)= -A\sp{-1}f_\infty=:u_\infty, \qquad \lim_{t\to\infty}u'(t)=0, \qquad \lim_{t\to\infty}A(t)u(t)=Au_\infty.$.