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Projectors on the Generalized Eigenspaces for Partial Differential Equations with Time Delay

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Infinite Dimensional Dynamical Systems

Part of the book series: Fields Institute Communications ((FIC,volume 64))

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 reaction-diffusion equation with delay and an age-structured model with delay.

Mathematics Subject Classification (2010): Primary 35K57, 34K15; Secondary 92D30

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Acknowledgements

Research of S. Ruan was partially supported by NSF grant DMS-1022728.

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Authors and Affiliations

  1. Institut de Mathématiques de Bordeaux UMR CNRS 5251, Université de Bordeaux, 3ter place de la victoire, Bat E 2 eme étage, 33076, Bordeaux, France

    Arnaut Ducrot & Pierre Magal

  2. Department of Mathematics, University of Miami, Coral Gables, FL, 33124-4250, USA

    Shigui Ruan

Authors
  1. Arnaut Ducrot
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  2. Pierre Magal
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  3. Shigui Ruan
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Correspondence to Arnaut Ducrot .

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Editors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, RI, USA

    John Mallet-Paret

  2. Department of Mathematics and Statistics, York University, Toronto, ON, Canada

    Jianhong Wu  & Huaiping Zhu  & 

  3. Georgia Institute of Technology, School of Mathematics, Atlanta, GA, USA

    Yingfei Yi

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Ducrot, A., Magal, P., Ruan, S. (2013). Projectors on the Generalized Eigenspaces for Partial Differential Equations with Time Delay. In: Mallet-Paret, J., Wu, J., Yi, Y., Zhu, H. (eds) Infinite Dimensional Dynamical Systems. Fields Institute Communications, vol 64. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4523-4_14

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