TY - CHAP

T1 - Projectors on the generalized eigenspaces for partial differential equations with time delay

AU - Ducrot, Arnaut

AU - Magal, Pierre

AU - Ruan, Shigui

N1 - Funding Information:
Research of S. Ruan was partially supported by NSF grant DMS-1022728.

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

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U2 - 10.1007/978-1-4614-4523-4_14

DO - 10.1007/978-1-4614-4523-4_14

M3 - Chapter

AN - SCOPUS:84874370281

SN - 9781461445227

T3 - Fields Institute Communications

SP - 353

EP - 390

BT - Infinite Dimensional Dynamical Systems

A2 - Mallet-Paret, John

ER -