Parabolic evolution equations in interpolation and extrapolation spaces (Q1108565)

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.