Differential and Integral Equations

Approximation and asymptotic behaviour of evolution families

Charles J. K. Batty and Ralph Chill

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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

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

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34G10: Linear equations [See also 47D06, 47D09]
Secondary: 34D05: Asymptotic properties 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]


Batty, Charles J. K.; Chill, Ralph. Approximation and asymptotic behaviour of evolution families. Differential Integral Equations 15 (2002), no. 4, 477--512. https://projecteuclid.org/euclid.die/1356060845

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