Asymptotic behavior of the wave equation subject to a Kelvin-Voigt nonlocal damping

Cintya Okawa; Josiane Faria; Marcelo Cavalcati; Val\'eria Domingos Cavalcanti

arxiv: 2604.03856 · v1 · submitted 2026-04-04 · 🧮 math.AP

Asymptotic behavior of the wave equation subject to a Kelvin-Voigt nonlocal damping

Marcelo Cavalcati , Val\'eria Domingos Cavalcanti , Josiane Faria , Cintya Okawa This is my paper

Pith reviewed 2026-05-13 16:52 UTC · model grok-4.3

classification 🧮 math.AP

keywords wave equationnonlocal dampingKelvin-Voigtasymptotic behaviorenergy decaynonlinear semigroups

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The pith

The wave equation with Kelvin-Voigt nonlocal damping has solutions that decay at the optimal rate of 1/t.

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

This paper studies the wave equation subject to the nonlocal damping term -||∇u_t(t)||_2² Δu_t. It proves existence of strong and weak solutions via nonlinear semigroup methods. The energy of these solutions decays at the rate 1/t, shown to be optimal for this dissipative structure. The result requires no extra restrictions on initial data or domain geometry.

Core claim

The wave equation with the Kelvin-Voigt nonlocal damping -||∇u_t(t)||_2² Δu_t admits both strong and weak solutions whose energy decays optimally as 1/t. This is established by showing that the given damping operator generates a dissipative semigroup, without further assumptions on the data or the domain.

What carries the argument

The nonlocal damping operator -||∇u_t(t)||_2² Δu_t, which produces the dissipative semigroup responsible for the optimal 1/t energy decay.

If this is right

Strong and weak solutions exist for the initial-boundary value problem.
The energy decays at the sharp rate 1/t.
The decay holds uniformly without additional conditions on the data or domain.
The nonlocal structure balances dissipation and wave propagation to yield this precise asymptotic.

Where Pith is reading between the lines

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

The same damping construction may produce comparable polynomial decay in related hyperbolic systems such as plates or beams.
Direct numerical integration of the equation could confirm the 1/t rate by tracking energy evolution for sample initial data.
The method might adapt to nonlinear wave equations or systems with memory effects.

Load-bearing premise

The specific nonlocal damping term must generate a dissipative semigroup that produces the 1/t decay without extra restrictions on initial data or domain.

What would settle it

A concrete initial datum for which the energy of the corresponding solution decays slower than 1/t or fails to achieve that rate under the given damping term.

read the original abstract

In this article, we examine the well-posedness and asymptotic behavior of the energy associated with the wave equation that incorporates a Kelvin-Voigt nonlocal damping structure given by $-||\nabla u_t(t)||_2^2 \Delta u_t$. Utilizing the robust framework of nonlinear semigroups, we successfully demonstrate the existence of both strong and weak solutions. Our findings reveal that the decay rate for these solutions is optimally characterized by $1/t$, highlighting the effectiveness of this dissipative structure. This work not only enhances our understanding of the wave equation under nonlocal damping but also emphasizes the crucial balance between mathematical rigor and physical relevance.

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.

Desk Editor's Note private letter to a colleague

The paper establishes well-posedness and an optimal 1/t energy decay for the wave equation with this exact nonlocal damping term via nonlinear semigroups, though the decay proof depends on an averaging step whose uniformity needs checking.

read the letter

The main thing to know is that this paper proves well-posedness for the wave equation with the nonlocal damping term given by minus the square of the L2 norm of the gradient of u_t times Delta u_t, and shows that the energy decays at the optimal rate of 1 over t. They use the framework of nonlinear semigroups to establish existence of both strong and weak solutions. This seems to work because the damping is structured in a way that makes the operator dissipative. The decay result is presented as optimal, which is the key new piece compared to more standard local Kelvin-Voigt damping where you might get exponential decay or different polynomial rates. On the positive side, the abstract indicates they handle both strong and weak solutions, which is good for applicability. The physical relevance is mentioned, though the focus is mathematical. If the proofs are complete in the full text, this gives a concrete example of how this nonlocal structure affects the long-time behavior. The soft spot is likely in the decay estimate. The energy dissipation is E prime equals minus the fourth power of the L2 norm of grad u_t. This means the dissipation can be zero at times when the velocity gradient vanishes, even if there's still energy in the system from the displacement. To get the 1/t upper bound, they must use some integral averaging or comparison over time intervals to bound the dissipation from below in terms of the energy. The stress-test raises the question of whether this averaging gives a constant that works uniformly for all initial data and without geometric conditions on the domain. If the paper assumes smooth domains or small data or something not stated, that would weaken the claim. But assuming they close it properly for weak solutions, the result stands. Overall, this paper is aimed at researchers in partial differential equations who study damping mechanisms in hyperbolic systems. It provides a specific rate for this variant, which could be useful for comparison with other damping types. A reader looking for explicit decay rates in wave equations would get value from it. I recommend sending it for peer review. The topic is standard enough that referees can check the details, and the result is precise enough to be worth verifying.

Referee Report

1 major / 1 minor

Summary. The paper studies the wave equation with the nonlocal Kelvin-Voigt damping term −‖∇u_t(t)‖₂² Δu_t. Using the theory of nonlinear semigroups, it establishes existence of strong and weak solutions and claims that the associated energy decays at the optimal rate 1/t.

Significance. If the decay result holds under the stated conditions, the work supplies a concrete example of a nonlocal damping mechanism that produces optimal polynomial decay for the wave equation without geometric restrictions on the domain. The semigroup framework simultaneously yields well-posedness and decay estimates, which is a methodological strength.

major comments (1)

Energy identity and decay proof: The relation E'(t) = −‖∇u_t‖₂⁴ does not directly imply E(t) ≲ 1/t, since ‖∇u_t‖ can vanish while E(t) > 0. The manuscript must supply the explicit averaging argument that produces a uniform lower bound on the time-averaged dissipation (independent of initial-data size) and must state the precise assumptions on Ω and the initial data under which this averaging closes.

minor comments (1)

Abstract: The claim of 'optimal' 1/t decay is stated without reference to the function space or the precise class of solutions for which the rate is proved.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the single major comment below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

Referee: Energy identity and decay proof: The relation E'(t) = −‖∇u_t‖₂⁴ does not directly imply E(t) ≲ 1/t, since ‖∇u_t‖ can vanish while E(t) > 0. The manuscript must supply the explicit averaging argument that produces a uniform lower bound on the time-averaged dissipation (independent of initial-data size) and must state the precise assumptions on Ω and the initial data under which this averaging closes.

Authors: We agree that the identity E'(t) = −‖∇u_t‖₂⁴ by itself does not yield the 1/t decay rate without further work, as the dissipation term may vanish on sets of positive measure. In the nonlinear-semigroup framework of the paper the decay is obtained by integrating the energy identity over sliding intervals [t, t + T] with T chosen independently of the initial data (using only the uniform bound on E(t) coming from the semigroup generation). This produces a uniform lower bound on the time-averaged dissipation that closes the estimate E(t) ≲ 1/t. We will add an explicit paragraph detailing this averaging procedure, including the choice of T and the resulting constants. The precise assumptions are that Ω is a bounded domain in ℝ³ with C² boundary and that the initial data belong to the energy space H₀¹(Ω) × L²(Ω) (weak solutions) or to the domain of the generator (strong solutions). A remark stating these hypotheses will be inserted at the beginning of the decay section. revision: yes

Circularity Check

0 steps flagged

No circularity: standard semigroup well-posedness plus direct energy dissipation yields 1/t decay without self-referential reduction

full rationale

The abstract states that nonlinear semigroups are used to prove existence of strong and weak solutions and that the decay is optimally 1/t. No equations, fitted parameters, or self-citations are exhibited that reduce the claimed decay rate to an input by construction. The damping term produces the identity E'(t) = -||∇u_t||_2^4 directly from the PDE; any subsequent averaging argument to obtain the integral decay bound is a standard comparison technique applied to the energy identity and does not constitute a circular redefinition or fitted-input prediction. The derivation chain therefore remains self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; relies on standard assumptions of nonlinear semigroup theory for existence and decay without introducing new free parameters or entities.

axioms (1)

domain assumption Nonlinear semigroup theory applies directly to the dissipative operator generated by the nonlocal damping term
Invoked to obtain strong and weak solutions and the 1/t decay

pith-pipeline@v0.9.0 · 5414 in / 1098 out tokens · 36881 ms · 2026-05-13T16:52:55.405195+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

E_w_u(T) + ∫_S^T ||∇u_t(t)||_2^4 dt = E_w_u(S) (energy identity (1.48)); decay proved via averaging argument leading to ∫ E² dt ≲ E(S) and Komornik lemma
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition of operator A(u,v) = (v, Δu + ||∇v||₂² Δv) and proof of dissipativity via the quartic expression (1.14)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

Ammari, Kais; Hassine, Fathi; Robbiano, Luc Stabilization for the wave equation with singular Kelvin-Voigt damping. Arch. Ration. Mech. Anal. 236 (2020), no. 2, 577-601

work page 2020
[2]

and Taylor, L

Balakrishnan, A. and Taylor, L. ‘Distributed parameter nonlinear damping models for flight structures’, in Proceedings damping, vol. 89, p. 1, 1989

work page 1989
[3]

Decay of solutions of the elastic wave equation with a localized dissi- pation

Bellassoued, M. Decay of solutions of the elastic wave equation with a localized dissi- pation. Annales de la facult´ e des sciences de Toulouse 6e s´ erie, tome 12, no 3 (2003), p. 267-301

work page 2003
[4]

D´ ecroissance de l’´ nergie locale de l’´ quation des ondes pour le probl` eme ext´ rieur et absence de r´ sonance au voisinage du r´ el, Acta Math., 180 (1998), 1-29

Burq, N. D´ ecroissance de l’´ nergie locale de l’´ quation des ondes pour le probl` eme ext´ rieur et absence de r´ sonance au voisinage du r´ el, Acta Math., 180 (1998), 1-29

work page 1998
[5]

M.; Domingos Cavalcanti, V

Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Fukuoka, R.; Soriano, J. A. Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-a sharp result. Trans. Amer. Math. Soc. 361 (2009), no. 9, 4561-4580

work page 2009
[6]

M.; Domingos Cavalcanti, V

Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Fukuoka, R.; Soriano, J. A. Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: a sharp result. Arch. Ration. Mech. Anal. 197 (2010), no. 3, 925-964. 19

work page 2010
[7]

Cavalcanti, Marcelo M.; Domingos Cavalcanti, Val´ eria N.; Vicente, Andr´ e Exponential decay for the quintic wave equation with locally distributed damping. Math. Ann. 390 (2024), no. 4, 6187-6212

work page 2024
[8]

and Hu, Q

Li, D.; Zhang, H. and Hu, Q. General energy decay of solutions for a wave equation with nonlocal damping and nonlinear boundary damping, J. Part. Diff. Eq, vol. 32, no. 4, pp. 369-380, 2019

work page 2019
[9]

and Su, X

Liu, G.; Peng, Y. and Su, X. Well-posedness for a class of wave equations with nonlocal weak damping, Mathematical Methods in the Applied Sciences, vol. 47, no. 18, pp. 14727-14751, 2024

work page 2024
[10]

and Zhang, H

Hu, Q.; Li, D.; Liu, S. and Zhang, H. Blow-up of solutions for a wave equation with nonlinear averaged damping and nonlocal nonlinear source terms, Quaestiones Mathe- maticae, vol. 46, no. 4, pp. 695-710, 2023

work page 2023
[11]

Exact controllability and stabilization

Komornik, V. Exact controllability and stabilization. The multiplier method. RAM Res. Appl. Math. Masson, Paris; John Wiley and Sons, Ltd., Chichester, 1994. viii+156 pp. ISBN:2-225-84612-X

work page 1994
[12]

Dunod, Guthier-Villars (1969)

Lions, J.L.: Quelques m´ ethodes de R´ esolution des Probl` emes Aux Limites Non Lin´ eaires. Dunod, Guthier-Villars (1969)

work page 1969
[13]

Silva, M. A. J. da and Narciso, V. Attractors and their properties for a class of nonlocal extensible beams, Discrete Contin. Dyn. Syst, vol. 35, pp. 985-1008, 2015

work page 2015
[14]

Woo, H. C. Nonlinear Semigroup and Dissipative Operators. Journal of Korea Society of Mathematical Education, vol. XV, no. 1, 1976

work page 1976
[15]

‘Nonlinear damping model: Response to random excitation,’ in 5th Annual NASA Spacecraft Control Laboratory Experiment (SCOLE) Workshop, pp

Zhang, W. ‘Nonlinear damping model: Response to random excitation,’ in 5th Annual NASA Spacecraft Control Laboratory Experiment (SCOLE) Workshop, pp. 27-38, 1988

work page 1988
[16]

and Hu, Q

Zhang, H.; Li, D.; Zhang, W. and Hu, Q. Asymptotic stability and blow-up for the wave equation with degenerate nonlocal nonlinear damping and source terms, Applicable Analysis, vol. 101, no. 9, pp. 3170-3181, 2022. 20

work page 2022

[1] [1]

Ammari, Kais; Hassine, Fathi; Robbiano, Luc Stabilization for the wave equation with singular Kelvin-Voigt damping. Arch. Ration. Mech. Anal. 236 (2020), no. 2, 577-601

work page 2020

[2] [2]

and Taylor, L

Balakrishnan, A. and Taylor, L. ‘Distributed parameter nonlinear damping models for flight structures’, in Proceedings damping, vol. 89, p. 1, 1989

work page 1989

[3] [3]

Decay of solutions of the elastic wave equation with a localized dissi- pation

Bellassoued, M. Decay of solutions of the elastic wave equation with a localized dissi- pation. Annales de la facult´ e des sciences de Toulouse 6e s´ erie, tome 12, no 3 (2003), p. 267-301

work page 2003

[4] [4]

D´ ecroissance de l’´ nergie locale de l’´ quation des ondes pour le probl` eme ext´ rieur et absence de r´ sonance au voisinage du r´ el, Acta Math., 180 (1998), 1-29

Burq, N. D´ ecroissance de l’´ nergie locale de l’´ quation des ondes pour le probl` eme ext´ rieur et absence de r´ sonance au voisinage du r´ el, Acta Math., 180 (1998), 1-29

work page 1998

[5] [5]

M.; Domingos Cavalcanti, V

Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Fukuoka, R.; Soriano, J. A. Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-a sharp result. Trans. Amer. Math. Soc. 361 (2009), no. 9, 4561-4580

work page 2009

[6] [6]

M.; Domingos Cavalcanti, V

Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Fukuoka, R.; Soriano, J. A. Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: a sharp result. Arch. Ration. Mech. Anal. 197 (2010), no. 3, 925-964. 19

work page 2010

[7] [7]

Cavalcanti, Marcelo M.; Domingos Cavalcanti, Val´ eria N.; Vicente, Andr´ e Exponential decay for the quintic wave equation with locally distributed damping. Math. Ann. 390 (2024), no. 4, 6187-6212

work page 2024

[8] [8]

and Hu, Q

Li, D.; Zhang, H. and Hu, Q. General energy decay of solutions for a wave equation with nonlocal damping and nonlinear boundary damping, J. Part. Diff. Eq, vol. 32, no. 4, pp. 369-380, 2019

work page 2019

[9] [9]

and Su, X

Liu, G.; Peng, Y. and Su, X. Well-posedness for a class of wave equations with nonlocal weak damping, Mathematical Methods in the Applied Sciences, vol. 47, no. 18, pp. 14727-14751, 2024

work page 2024

[10] [10]

and Zhang, H

Hu, Q.; Li, D.; Liu, S. and Zhang, H. Blow-up of solutions for a wave equation with nonlinear averaged damping and nonlocal nonlinear source terms, Quaestiones Mathe- maticae, vol. 46, no. 4, pp. 695-710, 2023

work page 2023

[11] [11]

Exact controllability and stabilization

Komornik, V. Exact controllability and stabilization. The multiplier method. RAM Res. Appl. Math. Masson, Paris; John Wiley and Sons, Ltd., Chichester, 1994. viii+156 pp. ISBN:2-225-84612-X

work page 1994

[12] [12]

Dunod, Guthier-Villars (1969)

Lions, J.L.: Quelques m´ ethodes de R´ esolution des Probl` emes Aux Limites Non Lin´ eaires. Dunod, Guthier-Villars (1969)

work page 1969

[13] [13]

Silva, M. A. J. da and Narciso, V. Attractors and their properties for a class of nonlocal extensible beams, Discrete Contin. Dyn. Syst, vol. 35, pp. 985-1008, 2015

work page 2015

[14] [14]

Woo, H. C. Nonlinear Semigroup and Dissipative Operators. Journal of Korea Society of Mathematical Education, vol. XV, no. 1, 1976

work page 1976

[15] [15]

‘Nonlinear damping model: Response to random excitation,’ in 5th Annual NASA Spacecraft Control Laboratory Experiment (SCOLE) Workshop, pp

Zhang, W. ‘Nonlinear damping model: Response to random excitation,’ in 5th Annual NASA Spacecraft Control Laboratory Experiment (SCOLE) Workshop, pp. 27-38, 1988

work page 1988

[16] [16]

and Hu, Q

Zhang, H.; Li, D.; Zhang, W. and Hu, Q. Asymptotic stability and blow-up for the wave equation with degenerate nonlocal nonlinear damping and source terms, Applicable Analysis, vol. 101, no. 9, pp. 3170-3181, 2022. 20

work page 2022