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Parabolic evolution equations in interpolation and extrapolation spaces. (English) Zbl 0654.47019
This paper deals with the Cauchy problem \(\dot u+A(t)u=f(t)\) in a Banach space \(X_ 0\), where A is a linear operator with time variable domain D(A(t)).
In the first part it is assumed the existence of Banach spaces \(X_ 1\) and \(X_{\nu}\) such that \(D(A(t))=X_ 1\subset X_{\nu}\subset X_ 0\) and A has some properties related to \(X_{\nu}\). One proves the existence for the Cauchy problem associated with \(A_{\nu}\)- the maximal restriction of A(t) to \(X_{\nu}.\)
Conversely, in Part II one presents a general method for the construction of “extrapolation spaces” and “extrapolated operators” which are semigroup generators. So, starting with the Cauchy problem for \(A_{\nu}\) one can find \(X_ 0\) and A(t) for which the same problem is well-posed.
The paper improves previous results of the author. He gives a strong motivation of the study and carefully establishes the relations with the references.
Reviewer: C.Marinov

MSC:
47D03 Groups and semigroups of linear operators
46M35 Abstract interpolation of topological vector spaces
47F05 General theory of partial differential operators
47E05 General theory of ordinary differential operators
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