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

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.