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Evolution Semigroups and Product Formulas for Nonautonomous Cauchy Problems
Author(s) -
Nickel Gregor
Publication year - 2000
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/(sici)1522-2616(200004)212:1<101::aid-mana101>3.0.co;2-3
Subject(s) - mathematics , banach space , product (mathematics) , pure mathematics , section (typography) , initial value problem , cauchy distribution , cauchy problem , space (punctuation) , product topology , mathematical analysis , geometry , linguistics , philosophy , advertising , business
In this paper, we study nonautonomous Cauchy problems ( NCP ) { u̇ ( t ) = A ( t ) u ( t ) u ( s ) = x ∈ X for a family of linear operators ( A ( t )) t ∈ I on some Banach space X by means of evolution semigroups. In particular, we characterize “stability” in the so called “hyperbolic case” on the level of evolution semigroups and derive a product formula for the solutions of ( NCP ). Moreover, in Section 4 we connect the “hyperbolic” and the “parabolic” case by showing, that integrals ∫ t s A (τ) d τ always define generators. This yields another product formula.