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Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models


About this Title

Pierre Magal, UMR CNRS 5251 IMB & INRIA, Sud-Ouest Anubis, Université of Bordeaux, 146 rue Léo Saignat, 33076 Bordeaux, France and Shigui Ruan, Department of Mathematics, University of Miami, Coral Gables, Florida 33124-4250

Publication: Memoirs of the American Mathematical Society
Publication Year: 2009; Volume 202, Number 951
ISBNs: 978-0-8218-4653-7 (print); 978-1-4704-0565-6 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-09-00568-7
Published electronically: July 22, 2009
MathSciNet review: 2559965
Keywords:Center manifold, semilinear Cauchy problem, non-dense domain, Hopf bifurcation, age structure model
MSC: Primary 37L10; Secondary 35K90, 35R20, 37L50, 92D25

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Integrated semigroups
  • Chapter 3. Spectral decomposition of the state space
  • Chapter 4. Center manifold theory
  • Chapter 5. Hopf bifurcation in age structured models

Abstract


Several types of differential equations, such as delay differential equations, age-structure models in population dynamics, evolution equations with boundary conditions, can be written as semilinear Cauchy problems with an operator which is not densely defined in its domain. The goal of this paper is to develop a center manifold theory for semilinear Cauchy problems with non-dense domain. Using Liapunov-Perron method and following the techniques of Vanderbauwhede et al. in treating infinite dimensional systems, we study the existence and smoothness of center manifolds for semilinear Cauchy problems with non-dense domain. As an application, we use the center manifold theorem to establish a Hopf bifurcation theorem for age structured models.

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