Projectors on the generalized eigenspaces for partial differential equations with time delay

Arnaut Ducrot, Pierre Magal, Shigui Ruan

Research output: Chapter in Book/Report/Conference proceedingChapter

8 Scopus citations

Abstract

To study the nonlinear dynamics, such as Hopf bifurcation, of partial differential equations with delay, one needs to consider the characteristic equation associated to the linearized equation and to determine the distribution of the eigenvalues; that is, to study the spectrum of the linear operator. In this paper we study the projectors on the generalized eigenspaces associated to some eigenvalues for linear partial differential equations with delay.We first rewrite partial differential equations with delay as non-densely defined semilinear Cauchy problems, then obtain formulas for the integrated solutions of the semilinear Cauchy problems with non-dense domain by using integrated semigroup theory, from which we finally derive explicit formulas for the projectors on the generalized eigenspaces associated to some eigenvalues. As examples, we apply the obtained results to study a reactiondiffusion equation with delay and an age-structured model with delay.

Original languageEnglish (US)
Title of host publicationInfinite Dimensional Dynamical Systems
EditorsJohn Mallet-Paret
Pages353-390
Number of pages38
DOIs
StatePublished - Mar 4 2013

Publication series

NameFields Institute Communications
Volume64
ISSN (Print)1069-5265

ASJC Scopus subject areas

  • Mathematics(all)

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  • Cite this

    Ducrot, A., Magal, P., & Ruan, S. (2013). Projectors on the generalized eigenspaces for partial differential equations with time delay. In J. Mallet-Paret (Ed.), Infinite Dimensional Dynamical Systems (pp. 353-390). (Fields Institute Communications; Vol. 64). https://doi.org/10.1007/978-1-4614-4523-4_14