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

Haozhe Shu

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.

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

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.

Paper Structure

This paper contains 16 sections, 25 theorems, 187 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main assumptions
  3. Main results
  4. Outline
  5. Spectral theory on Hilbert space
  6. Spectral properties of the infinitesimal generator $\mathcal{A}$
  7. Generalized spectral theory on Gelfand triple
  8. Gelfand triples
  9. Generalization of the characteristic equation
  10. Generalized retarded resolvent
  11. Properties of generalized spectrum
  12. Asymptotic behavior on Gelfand triple
  13. Linear stability of a coupled oscillator system with time delay
  14. Linearization at the incoherent state
  15. Obstacles caused by continuous spectrum
  16. ...and 1 more sections

Key Result

Theorem 1.1

Assume that $\kappa^{-1}K^\times A(\lambda)\kappa:X\to X$ is compact, uniformly in $\lambda$. We have $\hat{\sigma}(\mathcal{A})=\hat{\sigma}_p(\mathcal{A})$.

Figures (3)

  • Figure 1: In the classical sense, the contour cannot be deformed to the left complex half-plane due to existence of continuous spectrum (blue line). By extending the retarded resolvent via a Gelfand triple, continuous spectrum disappear from a new Riemann surface. Analytic contour deformation is applicable where resonance poles of $\mathcal{R_\tau}(\lambda)\kappa$ on the second Riemann sheet explains asymptotic behavior of weak solutions.
  • Figure 2: Linear stability region ($|k|<|k_c(\tau)|$) for the incoherent state with $\omega_0=\pi/2$.
  • Figure 3: Since the continuous spectrum $\sigma_c(\mathcal{A})=i\mathbb{R}$ disappear in the generalized sense, the integral contour can be deformed to the left half-plane (i.e. $\Gamma_1(\infty)\cup\sum_{p\in \mathbb{N}}\gamma_p$)

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1
  • Lemma 1: batkai1
  • Remark 1
  • Lemma 2: batkai1
  • Lemma 3
  • Definition 2
  • Remark 2
  • Remark 3
  • ...and 52 more