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Well-posedness of abstract Cauchy problems. (English) Zbl 0542.34053

In this paper a unified treatment of the Hille-Phillips-Fattorini results on well-posedness of the abstract Cauchy problem for \(u'(t)=Au(t)\) and \(u''(t)=Au(t)\) is presented. Namely, we give six equivalent definitions of well-posedness for the first and for the second order equation. The proofs of their equivalence to each other and to A being a semigroup or cosine family generator appear to be simpler than any of the other proofs in the literature. In the author’s dissertation (Tübingen, 1984) these results have been extended to the Cauchy problem for \[ u^{(n+1)}(t)- Au^{(n)}(t)-B_ 1u^{(n-1)}(t)-...-B_ nu(t)=0, \] where A is a semigroup generator and \(B_ 1,...,B_ n\) are closed operators whose domains contain the domain of A.

MSC:

34G10 Linear differential equations in abstract spaces
47D03 Groups and semigroups of linear operators
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References:

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