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Well-posedness and asymptotic behaviour of non-autonomous linear evolution equations. (English) Zbl 1044.34016
Lorenzi, Alfredo (ed.) et al., Evolution equations, semigroups and functional analysis. In memory of Brunello Terreni. Containing papers of the conference, Milano, Italy, September 27-28, 2000. Basel: BirkhĂ¤user (ISBN 3-7643-6791-1/hbk). Prog. Nonlinear Differ. Equ. Appl. 50, 311-338 (2002).
This paper is a survey article with 143 references – it appears that it will be an invaluable resource for beginners in this area and also those already working in this field. The topic is existence theory and asymptotic behavior for the nonautonomous linear evolution problem ${d\over dt} u(t) = A(t) u(t)+ f(t),\quad t\geq s,\;t,s\in J,\quad u(s) =x,$ in a Banach space $$X$$, where $$A(t)$$ are linear operators, $$f\in L^1_{\text{loc}}(J,X)$$ and $$J$$ is a closed interval of $$\mathbb{R}$$, and primarily addresses how the concept of evolution semigroups relates to these questions. Section 2 overviews known existence results. Section 3 considers asymptotic behavior, in particular exponential dichotomy of evolution families, including examples which illustrate some of the difficulties encountered in studying these problems . Section 4 discusses the evolution semigroup approach to studying the well-posedness of nonautonomous Cauchy problems. Section 5 presents various characterizations of exponential dichotomy. Section 6 presents sufficient conditions on $$A(t)$$ which imply the existence of an exponential dichotomy for the homogeneous problem and makes use of the results in section 5. The author also gives a new proof based on the spectral theory of evolution semigroups for a stability result due to R. Datko. The author believes that the approach of evolution semigroups, begun by J. S. Howland, D. E. Evans and L. Paquet in the seventies, might help to unify the diverse existence results for nonautonomous equations.
For the entire collection see [Zbl 1004.00018].

##### MSC:
 34G10 Linear differential equations in abstract spaces 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations 34D09 Dichotomy, trichotomy of solutions to ordinary differential equations