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arxiv: 2603.26274 · v2 · pith:PDG5K45Enew · submitted 2026-03-27 · 🧮 math.AP

Optimal energy decay rates for Klein-Gordon equations with Kelvin-Voigt damping

Filippo Dell'Oro , Lassi Paunonen , David Seifert This is my paper

Pith reviewed 2026-05-25 06:38 UTC · model grok-4.3

classification 🧮 math.AP
keywords Klein-Gordon equationKelvin-Voigt dampingenergy decayC0-semigroupasymptotic stabilitypolynomial decayspectral pointsimaginary axis
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The pith

The energy of every solution to the one-dimensional Klein-Gordon equation with Kelvin-Voigt damping converges to zero, with an optimal polynomial rate for certain solutions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the long-time behaviour of solutions to a one-dimensional linear Klein-Gordon equation that includes Kelvin-Voigt damping. It establishes that the total energy of any solution tends to zero despite the generator of the governing semigroup having multiple spectral points on the imaginary axis. The work also identifies the sharpest polynomial decay rate that holds for a particular class of solutions. A reader would care because the result quantifies how this form of damping forces dissipation in a wave-like system that otherwise supports non-decaying modes.

Core claim

For the one-dimensional linear Klein-Gordon equation with Kelvin-Voigt damping, the energy of every possible solution converges to zero as time goes to infinity, and an optimal polynomial energy decay rate is provided for a certain class of solutions, even though the generator of the associated C0-semigroup has multiple spectral points on the imaginary axis.

What carries the argument

The C0-semigroup generated by the spatial operator of the damped Klein-Gordon equation, whose spectrum contains multiple points on the imaginary axis.

If this is right

  • Energy tends to zero for every initial datum in the state space.
  • Polynomial decay rates are optimal within a distinguished subclass of solutions.
  • The Kelvin-Voigt term produces asymptotic stability for the one-dimensional system.
  • The presence of imaginary spectral points does not prevent eventual energy loss.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same semigroup techniques might classify decay rates when the spatial domain is an interval with different boundary conditions.
  • Comparable polynomial rates could appear in other hyperbolic equations whose generators share the multiple-imaginary-point feature.
  • Numerical schemes for long-time simulation of such systems could be validated against the derived decay exponents.

Load-bearing premise

The generator of the associated semigroup has multiple spectral points on the imaginary axis.

What would settle it

Exhibiting even one solution whose energy remains bounded away from zero for all future times would disprove the convergence claim.

read the original abstract

We study the long-time behaviour of solutions to a one-dimensional linear Klein-Gordon equation with Kelvin-Voigt damping. One of the interesting features of the equation is that the generator of the associated $C_0$-semigroup has multiple spectral points on the imaginary axis. As our main result, we show that the energy of every possible solution converges to zero as time goes to infinity and, moreover, we provide an optimal polynomial energy decay rate for a certain class of solutions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper studies the long-time behaviour of solutions to a one-dimensional linear Klein-Gordon equation with Kelvin-Voigt damping. It notes that the generator of the associated C0-semigroup has multiple spectral points on the imaginary axis. The main result is that the energy of every solution converges to zero as time goes to infinity, and optimal polynomial energy decay rates are provided for a certain class of solutions.

Significance. If the results hold, this work would be significant in the field of partial differential equations, particularly for damped hyperbolic systems. Demonstrating strong stability despite spectral points on the imaginary axis, along with optimal decay rates, could provide new insights into the role of Kelvin-Voigt damping and challenge or extend existing stability criteria for such equations.

major comments (1)
  1. [Abstract and spectral analysis] The abstract asserts that energy(E(t)) → 0 for every solution while noting multiple spectral points on iℝ as a defining feature. By standard C0-semigroup theory, this requires σ_p(A) ∩ iℝ = ∅ (an eigenvector for eigenvalue iω would produce a solution with constant energy). The manuscript must explicitly verify in its spectral analysis that these points lie only in the continuous or residual spectrum, with a clear separation argument or theorem; without this, the strong stability claim for all solutions is unsupported.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and valuable feedback. We respond to the major comment as follows.

read point-by-point responses

  1. Referee: [Abstract and spectral analysis] The abstract asserts that energy(E(t)) → 0 for every solution while noting multiple spectral points on the imaginary axis. By standard C0-semigroup theory, this requires σ_p(A) ∩ iℝ = ∅ (an eigenvector for eigenvalue iω would produce a solution with constant energy). The manuscript must explicitly verify in its spectral analysis that these points lie only in the continuous or residual spectrum, with a clear separation argument or theorem; without this, the strong stability claim for all solutions is unsupported.

    Authors: We acknowledge the referee's point that the strong stability requires the absence of point spectrum on the imaginary axis. Our analysis shows that the spectral points on iℝ are not eigenvalues, as the resolvent equation has no non-trivial solutions in the domain for λ = iω. To address the request for explicit verification, we will add a proposition in the revised manuscript that rigorously separates the point spectrum and confirms these points are in the continuous spectrum, including the necessary arguments based on the damping term and boundary conditions. This will support the claim that E(t) → 0 for all solutions. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from PDE structure and spectral assumptions via standard semigroup methods

full rationale

The claimed energy decay to zero for all solutions and the optimal polynomial rates are obtained by analyzing the generator's spectrum (including points on iR) and applying resolvent estimates or frequency-domain methods to the Klein-Gordon equation with Kelvin-Voigt damping. No step equates a derived quantity to a fitted parameter or self-citation by construction; the spectral points on iR are input data whose nature (point vs. continuous spectrum) is examined directly rather than assumed to produce the decay. The argument is self-contained against the equation and semigroup theory without reduction to its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information on free parameters, axioms or invented entities is available from the abstract alone.

pith-pipeline@v0.9.0 · 5605 in / 1025 out tokens · 41753 ms · 2026-05-25T06:38:08.863200+00:00 · methodology

discussion (0)

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Reference graph

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