## Differential and Integral Equations

### 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.

#### Article information

Source
Differential Integral Equations, Volume 15, Number 4 (2002), 477-512.

Dates
First available in Project Euclid: 21 December 2012