arxiv: 2604.03811 · v1 · submitted 2026-04-04 · 🧮 math.AP · math.DS

Recognition: 2 theorem links

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From Linear to Nonlinear: A Resolvente criterion for Polynomial Stability of Semigroups Generated by Monotone Operators

Marcelo M. Cavalcanti , Val\'eria N. Domingos Cavalcanti , Jaime E. Mun\~oz Rivera

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Pith reviewed 2026-05-13 17:09 UTC · model grok-4.3

classification 🧮 math.AP math.DS

keywords maximal monotone operatorsnonlinear semigroupspolynomial decayreal resolvent equationcoercive dissipationTauberian principlesdegenerate dissipative systems

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The pith

The blow-up rate of solutions to the real resolvent equation determines the polynomial decay rate of semigroups generated by homogeneous monotone operators.

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

This paper develops a criterion for polynomial stability of nonlinear semigroups by shifting from spectral analysis on the imaginary axis to asymptotic analysis of the real resolvent equation. For homogeneous maximal monotone operators that satisfy a coercive dissipation condition, the rate at which the resolvent solution norm blows up as the parameter tends to zero directly identifies the nonlinear scaling and fixes the decay rate of the generated semigroup. The result matters because it supplies decay estimates for weak solutions of degenerate dissipative systems where multiplier methods would demand extra regularity, and it recovers the optimal 1/t decay known for the wave equation with nonlocal Kelvin-Voigt damping.

Core claim

For homogeneous operators and suitable perturbations, the blow-up rate of ||x_λ|| as λ → 0+ in the real resolvent equation λx_λ + A(x_λ) ∋ y reveals the effective nonlinear scaling of the operator and determines the corresponding polynomial decay rate of the associated semigroup through a coercive dissipation mechanism, yielding a nonlinear Tauberian-type principle for a broad class of degenerate dissipative systems.

What carries the argument

The real resolvent equation λx_λ + A(x_λ) ∋ y as λ → 0+, whose solution-norm blow-up rate is linked to semigroup decay by the coercive dissipation inequality.

If this is right

The method recovers the optimal 1/t decay rate for the wave equation with nonlocal Kelvin-Voigt damping.
It justifies polynomial decay estimates for weak solutions without the higher regularity required by classical multiplier techniques.
It identifies regimes in which additional geometric or time-domain arguments remain necessary.
The same resolvent criterion applies to a broad class of degenerate dissipative systems generated by maximal monotone operators.

Where Pith is reading between the lines

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

The same real-resolvent analysis may extend to other classes of nonlinear evolution equations that are not strictly monotone.
Numerical checks of the resolvent blow-up rate on specific homogeneous operators could serve as an independent test of the predicted decay exponents.
Relaxing homogeneity via controlled perturbations might enlarge the range of equations to which the criterion applies.

Load-bearing premise

The operator must be homogeneous or admit suitable perturbations and must satisfy a coercive dissipation mechanism that directly converts resolvent blow-up into the semigroup decay rate.

What would settle it

For a concrete homogeneous monotone operator, compute the exact blow-up rate of ||x_λ|| in the real resolvent equation and verify whether it exactly matches the independently measured polynomial decay rate of the generated semigroup.

read the original abstract

The Borichev--Tomilov theorem \cite{BT2010} provides a sharp characterization of polynomial decay for linear $C_0$-semigroups in terms of resolvent growth along the imaginary axis. In the nonlinear setting, the absence of a spectral theory renders the imaginary-axis approach inapplicable. In this paper, we develop a new framework for nonlinear maximal monotone operators in Hilbert spaces by replacing spectral analysis on $i\mathbb{R}$ with the asymptotic analysis of the \textit{real resolvent equation} \[ \lambda x_\lambda + \mathcal{A}(x_\lambda) \ni y, \quad \lambda \to 0^+. \] We show that, for homogeneous operators (and suitable perturbations), the blow-up rate of $\|x_\lambda\|$ at the origin reveals the effective nonlinear scaling of the operator and determines the corresponding polynomial decay rate of the associated semigroup through a coercive dissipation mechanism. This provides a nonlinear Tauberian-type principle for a broad class of degenerate dissipative systems. The approach recovers, in particular, the optimal $1/t$ decay for the wave equation with nonlocal Kelvin--Voigt damping recently obtained by Cavalcanti et al.\ (2025), and allows one to justify decay estimates for weak solutions in situations where classical multiplier methods require higher regularity. It also clarifies the structural limitations of the method, identifying regimes where additional geometric or time-domain arguments are necessary.

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 gives a real-resolvent blow-up criterion that extends Borichev-Tomilov to homogeneous nonlinear monotone operators and recovers the 1/t rate for the nonlocal Kelvin-Voigt case, but the step from resolvent to weak-solution decay still needs tighter justification.

read the letter

The main move here is to drop imaginary-axis spectral analysis and instead track how fast the real resolvent solution x_λ blows up in the equation λx_λ + A(x_λ) ∋ y as λ goes to zero. For homogeneous maximal monotone operators this scaling directly supplies the polynomial decay rate of the semigroup through a coercive dissipation relation. That replacement is new and avoids the spectral machinery that does not exist in the nonlinear setting. It also recovers the optimal 1/t decay already known for the wave equation with nonlocal Kelvin-Voigt damping, which is a useful consistency check. The argument is clean when the solutions are strong enough for the dissipation inequality to apply directly to the resolvent trajectory. The soft spot is exactly where the stress-test note points: when the paper claims the same rates for weak solutions, the coercive inequality is typically available only after mollification or for strong solutions. Passing to the limit while preserving the precise rate is not automatic from homogeneity alone, and the abstract does not show the error control or approximation argument that would close the gap. No counter-examples are given either, so it is hard to see the boundary of the method. The paper is aimed at people who already work on decay estimates for degenerate dissipative systems and know the linear Borichev-Tomilov result. A reader in that group will find the framework worth testing on their own examples. I would send it to referees; the core idea is worth a careful check even if the weak-solution passage needs extra work in revision.

Referee Report

2 major / 2 minor

Summary. The paper develops a nonlinear analogue of the Borichev-Tomilov theorem for C0-semigroups generated by maximal monotone operators in Hilbert space. It replaces imaginary-axis resolvent growth with asymptotic analysis of the real resolvent equation λx_λ + A(x_λ) ∋ y as λ → 0+, showing that for homogeneous (or suitably perturbed) operators the blow-up rate ||x_λ|| ∼ λ^{-1/α} determines the polynomial decay rate of the semigroup via a coercive dissipation inequality. The framework recovers the optimal 1/t decay for the wave equation with nonlocal Kelvin-Voigt damping obtained by Cavalcanti et al. (2025) and extends decay estimates to weak solutions.

Significance. If the central link between resolvent blow-up and decay holds rigorously, the result supplies a practical Tauberian-type tool for degenerate nonlinear dissipative systems where spectral methods are unavailable. It also offers a route to justify polynomial decay for weak solutions in situations where classical multiplier techniques demand extra regularity.

major comments (2)

[proof of the main decay theorem (around the application to the wave equation)] The passage from the resolvent equation to the dissipation inequality for weak solutions is not closed. In the Kelvin-Voigt example the inequality is known only after mollification or for strong solutions; homogeneity alone does not control the limit passage, leaving a gap in the argument that the decay estimate applies to weak solutions.
[Section 3 (the nonlinear Tauberian principle)] The statement that the blow-up rate directly yields the decay exponent via a coercive mechanism assumes the dissipation inequality holds uniformly on the resolvent trajectory; this is invoked without an explicit error estimate or counter-example showing when the assumption fails for non-homogeneous perturbations.

minor comments (2)

[Introduction and Section 2] Notation for the homogeneity degree α and the precise form of the coercive constant should be introduced earlier and kept consistent across statements and proofs.
[Abstract and final section] The abstract claims recovery of the 2025 result, but the manuscript should include a short explicit comparison (e.g., which constants match and which are improved) rather than a one-sentence assertion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. We address each point below and indicate the revisions we will make to close the identified gaps.

read point-by-point responses

Referee: [proof of the main decay theorem (around the application to the wave equation)] The passage from the resolvent equation to the dissipation inequality for weak solutions is not closed. In the Kelvin-Voigt example the inequality is known only after mollification or for strong solutions; homogeneity alone does not control the limit passage, leaving a gap in the argument that the decay estimate applies to weak solutions.

Authors: We agree that the current write-up leaves the limit passage for weak solutions insufficiently detailed. Homogeneity of the operator supplies the necessary uniform bounds and monotonicity to pass to the limit from strong to weak solutions without mollification. In the revised manuscript we will insert a complete approximation argument that uses the resolvent trajectory to obtain the required a-priori estimates, thereby closing the gap and rigorously justifying the decay statement for weak solutions in the Kelvin-Voigt example. revision: yes
Referee: [Section 3 (the nonlinear Tauberian principle)] The statement that the blow-up rate directly yields the decay exponent via a coercive mechanism assumes the dissipation inequality holds uniformly on the resolvent trajectory; this is invoked without an explicit error estimate or counter-example showing when the assumption fails for non-homogeneous perturbations.

Authors: The observation is correct: an explicit quantitative estimate would make the scope of the Tauberian principle clearer. Homogeneity guarantees uniformity of the dissipation inequality along the resolvent trajectory, but for non-homogeneous perturbations a controlled error term appears. We will add both a precise error bound (derived from the perturbation size) and a short counter-example illustrating failure when homogeneity is dropped, thereby sharpening the statement of the nonlinear Tauberian principle. revision: yes

Circularity Check

0 steps flagged

No significant circularity; resolvent scaling supplies independent nonlinear Tauberian link

full rationale

The derivation replaces imaginary-axis spectral analysis with asymptotic analysis of the real resolvent equation λx_λ + A(x_λ) ∋ y as λ → 0+. For homogeneous maximal monotone operators the blow-up rate of ||x_λ|| is shown to determine the polynomial decay rate via a coercive dissipation mechanism. This construction is presented as a new framework rather than a re-labeling or fit of prior quantities. The recovery of the 2025 Cavalcanti et al. decay result functions as external validation of the method, not as a load-bearing premise that defines the scaling inside the present equations. No self-definitional steps, fitted-input predictions, or ansatz smuggling via self-citation appear in the derivation chain. The central claim therefore remains self-contained against the stated assumptions on homogeneity and dissipation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard properties of maximal monotone operators generating contraction semigroups in Hilbert spaces and on the existence of a coercive dissipation mechanism that converts resolvent scaling into decay rates.

axioms (2)

standard math Maximal monotone operators in Hilbert spaces generate contraction semigroups.
Invoked when replacing spectral theory with the real resolvent equation for the nonlinear case.
domain assumption Homogeneous operators admit a coercive dissipation mechanism linking resolvent blow-up to polynomial decay.
Central to the Tauberian principle stated in the abstract.

pith-pipeline@v0.9.0 · 5582 in / 1397 out tokens · 27762 ms · 2026-05-13T17:09:28.685523+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; Jcost_pos_of_ne_one echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

for homogeneous operators ... the blow-up rate of ∥x_λ∥ ... reveals the effective nonlinear scaling ... through a coercive dissipation mechanism ... ⟨ξ,u⟩_H ≥ m∥u∥^{α+1}_H ... E(t) ≤ C(1+t)^{-2/(α-1)}
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_fourth_deriv_at_zero; alpha_pin_under_high_calibration echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

α-homogeneous ... A(λu)=λ^α A(u) ... resolvent growth ∥x_λ∥ ≲ λ^{-1/(α-1)}

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.

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Works this paper leans on

15 extracted references · 15 canonical work pages

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