Approximation and asymptotic behaviour of evolution families. (Q1847665)

In this paper, the authors consider the long-time asymptotic behavior of non-autonomous Cauchy problems of the form: \(\frac{d}{dt}u(t)=A(t)u(t)\), \((t\geq s\geq 0)\), \(u(s)=x\) on a Banach space \(X\). Let \((A(t))_{t\geq 0}\) and \((B(t))_{t\geq 0}\) be two families of closed operators on \(X\), and let \((U(t,s))_{t\geq s\geq 0}\) and \((V(t,s))_{t\geq s\geq 0}\) be the associated evolution families. Then the authors obtain estimates for \(| | U(t,s)-V(t,s)| | \) in terms of \(| | A(\tau )^{-1}-B(\tau )^{-1}| | \) for \(s\leq \tau \leq t,\) in several different cases, namely when \(\{A(t)\}\) and \(\{B(t)\}\) satisfy the Acquistapace-Terreni conditions, or the Kato-Tanabe conditions, or the \(L^{p}\)-maximal regularity conditions for some \(1<p<\infty.\) Their results extend previous results by \textit{P. Acquistapace} and \textit{B. Terreni} [Rend. Semin. Mat. Univ. Padova 78, 47--107 (1987; Zbl 0646.34006)], by \textit{T. Kato} and \textit{H. Tanabe} [Osaka Math. J. 14, 107--133 (1962; Zbl 0106.09302)], by \textit{R. Schnaubelt} [J. Evol. Equ. 1, No. 1, 19--37 (2001; Zbl 1098.34551)] and by \textit{M. Hieber} and \textit{S. Monniaux} [Proc. Am. Math. Soc. 128, No. 4, 1047--1053 (2000; Zbl 0937.35195)].