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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

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
Full Text: DOI
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