Asymptotic behavior of solutions to linear evolution equations with time delay via a spectral theory on Gelfand triples

TL;DR

This paper develops a spectral theory on Gelfand triples to analyze the long-time behavior of linear evolution equations with time delay, revealing conditions for exponential decay and stability.

Contribution

It introduces a generalized spectral framework on Gelfand triples to handle continuous spectra obstructing long-time analysis of delayed equations.

Findings

Generalized spectrum consists of isolated eigenvalues under compactness.

Exponential decay is achieved via contour deformation in inverse Laplace transform.

Application to Kuramoto-Daido model shows linear stability of incoherent state.

Abstract

In this paper, a class of linear evolution equations with time delay is studied in which the presence of continuous spectrum on the imaginary axis obstructs the analysis of long-time dynamics. To address it, a generalized spectral framework on a Gelfand triple is utilized. When the spectral measure of the unperturbed term (a skew-adjoint operator) admits some analyticity condition, the resolvent is extended to a generalized resolvent. Called generalized spectrum, the collection of singularities on the Riemann surface of the generalized resolvent may differ from the spectrum in the usual sense because of the change of topology via the Gelfand triple. It is shown that under some compactness assumption, the generalized spectrum consists only of isolated generalized eigenvalues (resonance poles). This structure allows contour deformation in the inverse Laplace representation and yields exponential decay in a weak topology. As an application, we analyze the continuum limit of the Kuramoto-Daido model with time delay and prove linear stability of the incoherent state in the weak coupling regime.