Approximation and asymptotic behaviour of evolution families

Abstract

Let $(A(t))_{t \ge 0}$ and $(B(t))_{t \ge 0}$ be two families of closed operators satisfying the Acquistapace--Terreni conditions or the Kato--Tanabe conditions, or assumptions of maximal regularity, and let $(U(t,s))_{t > s \ge 0}$ and $(V(t,s))_{t > s\ge0}$ be the associated evolution families. We obtain some estimates for $ \| {U(t,s) - V(t,s)} \| $ in terms of $ \| {A(\tau)^{-1} - B(\tau)^{-1}} \| $ for $s \le \tau \le t$. We deduce some results showing that if $ \| {A(\tau)^{-1} - B(\tau)^{-1}} \| \to 0$ sufficiently quickly as $\tau \to \infty$, then $U$ and $V$ have similar asymptotic behaviour.