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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

$\left(1\right)\phantom{\rule{1.em}{0ex}}{u}^{\text{'}}\left(t\right)-A\left(t\right)u\left(t\right)=f\left(t\right),\phantom{\rule{1.em}{0ex}}t\in \left[0,T\right],\phantom{\rule{1.em}{0ex}}{u}^{\text{'}}\left(0\right)=x$

is studied in a Banach space E, with $x\in E$ and f:[0,T]$\to E$ as prescribed data. In (1) $\left\{$ A(t)$\right\}$ $t\in \left[0,T\right]$ is a family of closed linear operators in E which are generators of analytic semigroups, and whose domain ${D}_{A\left(t\right)}$ 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

##### MSC:
 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions