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Automatic Well‐Posedness with the Abstract Cauchy Problem on a Fréchet Space
Author(s) -
Delaubenfels Ralph
Publication year - 1993
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-48.3.526
Subject(s) - semigroup , mathematics , space (punctuation) , cauchy problem , equicontinuity , initial value problem , cauchy distribution , complete metric space , operator (biology) , pure mathematics , mathematical analysis , banach space , computer science , operating system , biochemistry , chemistry , repressor , transcription factor , gene
For an arbitrary closed linear operator, A , on a Fréchet space, we introduce what we shall call its solution space . This is a Fréchet space that contains all initial data for which the corresponding abstract Cauchy problem has a unique global mild solution. We show that A , restricted to this space, generates a locally equicontinuous strongly continuous semigroup. Corollaries include an almost immediate proof of the fundamental relationship between generating a strongly continuous semigroup and having a unique mild solution, for all initial data. More generally, we show how the solution space may be used to present a simplified and unified approach to different types of semigroups and their relationships to each other and the abstract Cauchy problem.