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A unified approach to abstract linear nonautonomous parabolic equations. (English) Zbl 0646.34006

The paper can be considered as a review, in which - on the basis of certain assumptions - a unified approach to abstract linear nonautonomous parabolic equations is proposed. In particular, the linear parabolic Cauchy problem

(1)u ' (t)-A(t)u(t)=f(t),t[0,T],u ' (0)=x

is studied in a Banach space E, with xE and f:[0,T]E as prescribed data. In (1) { A(t)} t[0,T] is a family of closed linear operators in E which are generators of analytic semigroups, and whose domain D A(t) may change with t and be not dense in E. Existence, uniqueness and regularity results are illustrated. The paper consists of 7 sections, rich in definitions, lemmas, propositions and theorems. In the last section examples and comparisons with the available literature are discussed.

Reviewer: V.C.Boffi

34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions