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Differential operators with non dense domain. (English) Zbl 0652.34069

The Cauchy problem in the Banach space E of the form \(u'(t)=Au(t)+f(t),\) \(t\in [0,T]\); \(u(0)=u_ 0\) where A:D\(\subseteq E\to E\) is a closed linear operator with non dense domain D, f: [0,T]\(\to E\), \(u_ 0\in E\), is considered. Analogues of well-known Hille-Yosida theorem (the assumption of the density of D in E is excepted) are proved. A more particular situation in which A generates an analytical semigroup (not necessarily strongly continuous at the origin) is investigated. As an application, the authors consider several examples of partial differential equations of hyperbolic and ultraparabolic type, and parabolic equations with infinitely many variables.
Reviewer: R.R.Akhmerov

MSC:

34G10 Linear differential equations in abstract spaces
47D03 Groups and semigroups of linear operators
35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
35L20 Initial-boundary value problems for second-order hyperbolic equations
35K99 Parabolic equations and parabolic systems
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