[Paper Review] Examples of very unstable linear partial functional differential equations

Р. С. Исмагилов, Н. А. Раутиан+1 more|arXiv (Cornell University)|Feb 17, 2014

Advanced Mathematical Modeling in Engineering4 references4 citations

TL;DR

This paper demonstrates that linear partial functional differential equations with delay in the Laplacian operator exhibit extreme instability, as their spectra contain eigenvalues with real parts tending to infinity. Using Fourier analysis and characteristic quasipolynomials, the authors prove that such equations—specifically a delayed heat equation, a delayed wave equation, and a perturbed delayed wave equation—possess unstable spectral structures, contrasting sharply with stable spectra of standard parabolic and hyperbolic equations.

ABSTRACT

We consider the examples of partial functional differential equations with delay in the Laplacian. First of these equations is linear parabolic equation, the second one is linear hyperbolic equation, third equation is perturbed hyperbolic equation with delay. We show that there are the sequence of eigenvalues in both cases with real parts tends to plus infinity.

Motivation & Objective

• To demonstrate that partial functional differential equations with delay in the Laplacian exhibit extreme spectral instability.
• To contrast the spectral behavior of time-delayed PDEs with that of classical parabolic and hyperbolic equations.
• To analyze the asymptotic distribution of eigenvalues for characteristic quasipolynomials arising from delayed PDEs.
• To show that standard weighted Sobolev spaces are insufficient for solving initial value problems of such delayed equations.
• To establish that the spectra of delayed equations can have eigenvalues with real parts diverging to +∞, indicating fundamental instability.

Proposed method

• Applying the Fourier method to decompose solutions into eigenfunction expansions in the spatial domain (0, π), reducing PDEs to infinite systems of ordinary delay differential equations.
• Deriving characteristic quasipolynomials for each equation: λ + n²e^(-λh) = 0 (parabolic), λ² + n²e^(-λh) = 0 (hyperbolic), and λ² + n²(1 + e^(-λh)) = 0 (perturbed hyperbolic).
• Using complex analysis and asymptotic analysis to study the location of roots of the characteristic quasipolynomials in the complex plane.
• Applying Lemma 1 to prove the existence of a unique solution λ(w) to λ + b·lnλ = w in a disk around w for large |w|, enabling asymptotic analysis of eigenvalues.
• Analyzing the asymptotic behavior of eigenvalues as n → ∞, showing that real parts grow logarithmically as ln(n²) − ln|π/2 + 2πk| + o(1/|k|).
• Comparing spectral structures of delayed equations with classical Maxwell-Cattaneo and undelayed equations to highlight qualitative differences in stability.

Experimental results

Research questions

• RQ1 Do linear partial functional differential equations with delay in the Laplacian exhibit spectral instability?
• RQ2 How does the spectrum of a delayed heat equation differ from that of a standard parabolic equation?
• RQ3 Can the eigenvalues of delayed hyperbolic-type equations have real parts tending to +∞?
• RQ4 Why do standard weighted Sobolev spaces fail to solve initial value problems for such delayed PDEs?
• RQ5 What is the asymptotic behavior of the eigenvalues of the characteristic quasipolynomials for delayed PDEs as n → ∞?

Key findings

• The characteristic quasipolynomial λ + n²e^(-λh) = 0 for the delayed heat equation has eigenvalues λ_nk with real parts satisfying Re(λ_nk) = ln(n²) − ln|π/2 + 2πk| + o(1/|k|), which can grow without bound as k → ∞.
• For the delayed wave equation λ² + n²e^(-λh) = 0, the real parts of eigenvalues also tend to +∞ along certain sequences, indicating extreme instability.
• The perturbed delayed wave equation λ² + n²(1 + e^(-λh)) = 0 has a stable spectrum in the sense that eigenvalues remain in a left half-plane {Re λ ≤ ω} for any ω > 0, contrasting with the other two equations.
• The asymptotic expansion of eigenvalues for the delayed heat equation shows that Re(λ_nk) ~ ln(n²) − ln|π/2 + 2πk|, which diverges as |k| increases.
• The spectrum of the delayed equation (3) is structurally different from the classical Maxwell-Cattaneo equation (1), with the former having eigenvalues spreading into the right half-plane.
• The failure of standard weighted Sobolev spaces W_{2,γ}^1(R_+, A²) to contain solutions is due to the analyticity of Laplace transforms in Re λ > γ, which cannot accommodate eigenvalues with Re λ → +∞.