# Existence Families, Functional Calculi and Evolution Equations

Part of the Lecture Notes in Mathematics book series (LNM, volume 1570)

Part of the Lecture Notes in Mathematics book series (LNM, volume 1570)

This book presents an operator-theoretic approach to ill-posed evolution equations. It presents the basic theory, and the more surprising examples, of generalizations of strongly continuous semigroups known as 'existent families' and 'regularized semigroups'. These families of operators may be used either to produce all initial data for which a solution in the original space exists, or to construct a maximal subspace on which the problem is well-posed. Regularized semigroups are also used to construct functional, or operational, calculi for unbounded operators. The book takes an intuitive and constructive approach by emphasizing the interaction between functional calculus constructions and evolution equations. One thinks of a semigroup generated by A as etA and thinks of a regularized semigroup generated by A as etA g(A), producing solutions of the abstract Cauchy problem for initial data in the image of g(A). Material that is scattered throughout numerous papers is brought together and presented in a fresh, organized way, together with a great deal of new material.

Semigroups of operators abstract Cauchy problem calculus evolution equations functional calculus maximum

- DOI https://doi.org/10.1007/BFb0073401
- Copyright Information Springer-Verlag Berlin Heidelberg 1994
- Publisher Name Springer, Berlin, Heidelberg
- eBook Packages Springer Book Archive
- Print ISBN 978-3-540-57703-4
- Online ISBN 978-3-540-48322-9
- Series Print ISSN 0075-8434
- Series Online ISSN 1617-9692
- About this book