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Evolution semigroups for nonautonomous Cauchy problems. (English) Zbl 0938.47030

Summary: We characterize wellposedness of nonautonomous, linear Cauchy problems \[ \begin{cases} \dot u(t)= A(t)u(t)\\ u(s)= x\in X\end{cases}\tag{NCP} \] on a Banach space \(X\) by the existence of certain evolution semigroups.
Then, we use these generation results for evolution semigroups to derive wellposedness for nonautonomous Cauchy problems under some “concrete” conditions. As a typical example, we discuss the so-called “parabolic” case.

MSC:

47D06 One-parameter semigroups and linear evolution equations
47A55 Perturbation theory of linear operators
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