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Theory and Applications of Abstract Semilinear Cauchy Problems

  • Pierre Magal
  • Shigui Ruan

Part of the Applied Mathematical Sciences book series (AMS, volume 201)

Table of contents

  1. Front Matter
    Pages i-xxii
  2. Pierre Magal, Shigui Ruan
    Pages 1-55
  3. Pierre Magal, Shigui Ruan
    Pages 57-99
  4. Pierre Magal, Shigui Ruan
    Pages 101-164
  5. Pierre Magal, Shigui Ruan
    Pages 165-216
  6. Pierre Magal, Shigui Ruan
    Pages 217-248
  7. Pierre Magal, Shigui Ruan
    Pages 249-308
  8. Pierre Magal, Shigui Ruan
    Pages 309-356
  9. Pierre Magal, Shigui Ruan
    Pages 357-449
  10. Pierre Magal, Shigui Ruan
    Pages 451-521
  11. Back Matter
    Pages 523-543

About this book

Introduction

Several types of differential equations, such as functional differential equation, age-structured models, transport equations, reaction-diffusion equations, and partial differential equations with delay, can be formulated as abstract Cauchy problems with non-dense domain. This monograph provides a self-contained and comprehensive presentation of the fundamental theory of non-densely defined semilinear Cauchy problems and their applications. Starting from the classical Hille-Yosida theorem, semigroup method, and spectral theory, this monograph introduces the abstract Cauchy problems with non-dense domain, integrated semigroups, the existence of integrated solutions, positivity of solutions, Lipschitz perturbation, differentiability of solutions with respect to the state variable, and time differentiability of solutions. Combining the functional analysis method and bifurcation approach in dynamical systems, then the nonlinear dynamics such as the stability of equilibria, center manifold theory, Hopf bifurcation, and normal form theory are established for abstract Cauchy problems with non-dense domain. Finally applications to functional differential equations, age-structured models, and parabolic equations are presented. This monograph will be very valuable for graduate students and researchers in the fields of abstract Cauchy problems, infinite dimensional dynamical systems, and their applications in biological, chemical, medical, and physical problems.


Keywords

abstract semilinear cauchy problems densely and non-densely defined cauchy problems integrated semigroup theory spectral theory of linear operators center manifold theory Hopf bifurcation normal form theory functional differential equations age-structured models parabolic equations

Authors and affiliations

  • Pierre Magal
    • 1
  • Shigui Ruan
    • 2
  1. 1.Institut de Mathématiques de BordeauxUniversité de BordeauxTalenceFrance
  2. 2.Department of MathematicsUniversity of MiamiCoral GablesUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-030-01506-0
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-01505-3
  • Online ISBN 978-3-030-01506-0
  • Series Print ISSN 0066-5452
  • Series Online ISSN 2196-968X
  • Buy this book on publisher's site