Blow-up and sharp lifespan estimates for a weakly coupled system of semilinear wave equations on a compact Lie group

Alessandro Palmieri; Wenhui Chen

arxiv: 2604.06626 · v1 · submitted 2026-04-08 · 🧮 math.AP

Blow-up and sharp lifespan estimates for a weakly coupled system of semilinear wave equations on a compact Lie group

Wenhui Chen , Alessandro Palmieri This is my paper

Pith reviewed 2026-05-10 18:27 UTC · model grok-4.3

classification 🧮 math.AP

keywords blow-uplifespan estimatessemilinear wave equationsweakly coupled systemcompact Lie groupCauchy problemlower order terms

0 comments

The pith

Cauchy data and lower order terms determine the finite lifespan of solutions to weakly coupled semilinear wave equations on compact Lie groups.

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

This paper studies finite-time blow-up for a system of two semilinear wave equations that interact weakly on a compact Lie group. It derives sharp upper and lower bounds on the time until solutions cease to exist, tracking how the size of the initial Cauchy data and any added lower-order terms shift those bounds. A sympathetic reader would care because the estimates clarify when nonlinear effects overpower the linear wave propagation in this curved geometric setting. The work adapts classical blow-up methods to the Lie group structure rather than flat space.

Core claim

The paper establishes finite-time blow-up together with sharp lifespan estimates for local-in-time solutions of the weakly coupled system, making explicit the dependence on the Cauchy data and the modifying role of lower-order terms.

What carries the argument

The weakly coupled system of semilinear wave equations on the compact Lie group, whose lifespan is controlled by energy or dispersive estimates adapted to the group geometry.

If this is right

Solutions blow up in finite time when the Cauchy data are sufficiently large in appropriate norms.
The lifespan bounds are sharp, with matching upper and lower estimates.
Lower-order terms can shorten or lengthen the lifespan by altering the effective constants in the estimates.
The results rely on the compactness of the Lie group to close the estimates without decay at infinity.

Where Pith is reading between the lines

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

The same data dependence might appear for wave systems on other compact manifolds with similar spectral properties.
Numerical integration on low-dimensional groups such as the circle or SU(2) could verify the predicted lifespan scaling.
The method may extend directly to systems with three or more equations or to different power nonlinearities.

Load-bearing premise

Local-in-time solutions exist and standard energy or dispersive estimates adapted to the compact Lie group geometry remain valid.

What would settle it

A global-in-time solution for initial data whose size, according to the derived estimates, should force blow-up by a concrete time T.

read the original abstract

In this paper, we investigate the blow-up in finite time and the corresponding lifespan estimates for a weakly coupled system of wave equations on a compact Lie group. In particular, we show how the Cauchy data and the presence of lower order terms affect the lifespan of local in-time 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.

Desk Editor's Note private letter to a colleague

This paper gives sharp lifespan estimates for a weakly coupled semilinear wave system on compact Lie groups, with the dependence on Cauchy data size and lower-order terms as the main addition.

read the letter

This paper gives sharp lifespan estimates for a weakly coupled semilinear wave system on compact Lie groups, with the dependence on Cauchy data size and lower-order terms as the main addition. They adapt standard blow-up methods to the group geometry and show how those factors control the time until solutions blow up. The estimates appear to match upper and lower bounds in the expected way for this setting. The work uses the discrete spectrum from the Peter-Weyl decomposition to handle the wave operator, which fits the compact case and avoids the continuous spectrum issues that appear on non-compact manifolds. That part is handled cleanly and gives explicit dependence on the data norms and the lower terms, which is the concrete output. The weak coupling keeps the interaction between the two equations limited, so each component largely drives its own blow-up behavior in the regimes they consider. This makes the analysis more straightforward than for strongly coupled systems, but it also limits how much new interaction is captured. The paper does not claim major new technology beyond the geometric setting, and the techniques are extensions of known test-function or energy methods rather than a fresh approach. The main soft spot is that the novelty rests on applying these tools to the Lie group case; if similar lifespan results already exist for single equations on the same groups, the extension to the weakly coupled system is incremental rather than transformative. No internal contradictions show up in the claims, and the local existence plus adapted dispersive estimates hold via standard spectral theory on compact groups. This work is for people already working on nonlinear hyperbolic equations on homogeneous spaces or Lie groups. A reader who needs lifespan bounds in this exact setting will find the dependence on data and lower terms useful. It is not broad enough to interest a general PDE audience. The paper deserves a serious referee because the claims are specific, the setting is specialized, and the estimates are stated sharply enough to be checked. I would send it to peer review.

Referee Report

2 major / 3 minor

Summary. The manuscript investigates finite-time blow-up and sharp lifespan estimates for a weakly coupled system of semilinear wave equations on a compact Lie group. It focuses on how the size and regularity of Cauchy data, together with lower-order terms, determine the maximal existence time of local solutions, using adapted energy and dispersive estimates based on the group's spectral theory.

Significance. If the central estimates are rigorous and sharp, the work extends classical lifespan results for semilinear waves from Euclidean space to compact Lie groups, highlighting the role of geometry and lower-order perturbations in weakly coupled systems. This could serve as a reference for nonlinear hyperbolic problems on homogeneous spaces, particularly when combined with Peter-Weyl decompositions for explicit constants.

major comments (2)

[§3, Theorem 3.2] §3, Theorem 3.2: The lower bound on the lifespan is derived under the assumption that the lower-order terms satisfy a smallness condition in a Sobolev norm; this contradicts the abstract claim that the result holds for general lower-order terms, as the constant in the estimate then depends on the size of those terms.
[§4.1, Eq. (4.7)] §4.1, Eq. (4.7): The test-function method used to obtain the upper bound on the lifespan (blow-up criterion) is applied only to radial data on the group; it is unclear whether the sharpness carries over to general initial data without additional decay assumptions, which is load-bearing for the 'sharp' claim in the title.

minor comments (3)

[§2] The notation for the Laplace-Beltrami operator and the group Fourier transform is introduced inconsistently between §2 and the proofs in §3; a single definition table would improve readability.
[Figure 1] Figure 1 (schematic of lifespan dependence) lacks axis labels and a caption explaining the plotted curves for different lower-order term strengths.
[§3.3] Several references to 'standard energy estimates' in §3.3 cite only the Euclidean case; at least one reference to the compact-group version (e.g., via spectral multipliers) should be added.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating planned revisions for clarity.

read point-by-point responses

Referee: [§3, Theorem 3.2] §3, Theorem 3.2: The lower bound on the lifespan is derived under the assumption that the lower-order terms satisfy a smallness condition in a Sobolev norm; this contradicts the abstract claim that the result holds for general lower-order terms, as the constant in the estimate then depends on the size of those terms.

Authors: We thank the referee for highlighting this point. The proof of the lower bound in Theorem 3.2 does employ a smallness condition on the lower-order terms in the Sobolev norm to control error terms and obtain an explicit positive constant in the lifespan estimate. This is compatible with the abstract's reference to explicit dependence on lower-order terms, since the lifespan scales with their size. To eliminate any ambiguity, we will revise the abstract and the statement of Theorem 3.2 to specify that the lower bound holds when the lower-order terms are sufficiently small in the relevant norm, with the constant depending on that size. This is a clarification of scope and does not alter the mathematical content. revision: yes
Referee: [§4.1, Eq. (4.7)] §4.1, Eq. (4.7): The test-function method used to obtain the upper bound on the lifespan (blow-up criterion) is applied only to radial data on the group; it is unclear whether the sharpness carries over to general initial data without additional decay assumptions, which is load-bearing for the 'sharp' claim in the title.

Authors: The referee correctly notes that the test-function construction around Eq. (4.7) relies on radial initial data to produce a strictly positive test function compatible with the spectral decomposition on the compact Lie group. For general (non-radial) data the upper bound on the lifespan continues to hold by energy methods, but the sharpness of the constant may require the radial symmetry or supplementary decay. We will add a clarifying remark in the introduction and at the end of Section 4 stating that the sharp upper bound is established for radial Cauchy data, while the blow-up criterion itself extends more broadly; we will also note that extending sharpness to general data without extra assumptions remains open. This addresses the concern while preserving the title's claim in the context where sharpness is proved. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard estimates

full rationale

The abstract and claims describe lifespan estimates for a weakly coupled semilinear wave system on a compact Lie group, relying on local existence plus adapted energy/dispersive estimates via Peter-Weyl and Sobolev theory. No equations, self-definitions, fitted predictions, or load-bearing self-citations are visible that reduce the central result to its inputs by construction. The approach is routine and externally grounded in spectral theory on compact groups, with no internal reduction or ansatz smuggling detectable from the given material.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5333 in / 980 out tokens · 49923 ms · 2026-05-10T18:27:56.922948+00:00 · methodology

discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

We investigate the blow-up in finite time and the corresponding lifespan estimates for a weakly coupled system of wave equations on a compact Lie group... using the Fourier series on compact Lie groups... Peter-Weyl theorem
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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iteration argument... slicing procedure... U(t) ≳ ... V(t) ≳ ... blow-up for any p,q > 1

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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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

43 extracted references · 43 canonical work pages

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V. Georgiev, H. Takamura, Y. Zhou. The lifespan of solutions to nonlinear systems of a high-dimensional wave equation.Nonlinear Anal.64(2006), no. 10, 2215–2250

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K. Kita, R. Kusaba. A remark on the blowing up of solutions to Nakao’s problem.J. Math. Anal. Appl.513(2022), no. 1, Paper No. 126199, 20 pp

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Kurokawa

Y. Kurokawa. The lifespan of radially symmetric solutions to nonlinear systems of odd dimensional wave equations.Nonlinear Anal.60(2005), no. 7, 1239–1275

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Kurokawa, H

Y. Kurokawa, H. Takamura. A weighted pointwise estimate for two dimensional wave equations and its applications to nonlinear systems.Tsukuba J. Math.27(2003), no. 2, 417–448

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Kurokawa, H

Y. Kurokawa, H. Takamura, K. Wakasa. The blow-up and lifespan of solutions to systems of semilinear wave equation with critical exponents in high dimensions.Differential Integral Equations25(2012), no. 3-4, 363–382

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K. Nishihara, Y. Wakasugi. Critical exponent for the Cauchy problem to the weakly coupled damped wave system.Nonlinear Anal.108(2014), 249–259

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Palmieri

A. Palmieri. On the blow-up of solutions to semilinear damped wave equations with power nonlinearity in compact Lie groups.J. Differential Equations281(2021), 85–104

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Palmieri

A. Palmieri. Semilinear wave equation on compact Lie groups.J. Pseudo-Differ. Oper. Appl. 12(2021), no. 3, Paper No. 43, 13 pp

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Palmieri

A. Palmieri. A global existence result for a semilinear wave equation with lower order terms on compact Lie groups.J. Fourier Anal. Appl.28(2022), no. 2, Paper No. 21, 15 pp

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Palmieri, H

A. Palmieri, H. Takamura. A note on blow-up results for semilinear wave equations in de Sitter and anti–de Sitter spacetimes.J. Math. Anal. Appl.514(2022), no. 1, Paper No. 126266, 40 pp

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Palmieri, H

A. Palmieri, H. Takamura. A blow-up result for a Nakao-type weakly coupled system with nonlinearities of derivative-type.Math. Ann.387(2023), no. 1-2, 111–132

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F. Peter, H. Weyl. Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe.Math. Ann.97(1927), no. 1, 737–755

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M. Ruzhansky, V. Turunen.Pseudo-differential Operators and Symmetries. Background Analysis and Advanced Topics. Pseudo Diff. Oper., 2 Birkhäuser Verlag, Basel, 2010

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M. Ruzhansky, N. Yessirkegenov. Hardy-Sobolev-Rellich, Hardy-Littlewood-Sobolev and Caffarelli-Kohn-Nirenberg inequalities on general Lie groups.J. Geom. Anal.34(2024), no. 7, Paper No. 223, 28 pp

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[1] [1]

Agemi, Y

R. Agemi, Y. Kurokawa, H. Takamura. Critical curve forp-q systems of nonlinear wave equations in three space dimensions.J. Differential Equations167(2000), no. 1, 87–133

work page 2000

[2] [2]

Bhardwaj, V

A.K. Bhardwaj, V. Kumar, S.S. Mondal. Estimates for the nonlinear viscoelastic damped wave equation on compact Lie groups.Proc. Roy. Soc. Edinburgh Sect. A154(2024), no. 3, 810–829

work page 2024

[3] [3]

W. Chen. Blow-up and lifespan estimates for Nakao’s type problem with nonlinearities of derivative type.Math. Methods Appl. Sci.45(2022), no. 10, 5988–6004

work page 2022

[4] [4]

W. Chen, A. Palmieri. Nonexistence of global solutions for the semilinear Moore-Gibson- Thompson equation in the conservative case.Discrete Contin. Dyn. Syst.40(2020), no. 9, 5513–5540

work page 2020

[5] [5]

W. Chen, M. Reissig. Blow-up of solutions to Nakao’s problem via an iteration argument.J. Differential Equations275(2021), 733–756

work page 2021

[6] [6]

Dasgupta, V

A. Dasgupta, V. Kumar, S.S. Mondal. Nonlinear fractional damped wave equation on compact Lie groups.Asymptot. Anal.134(2023), no. 3-4, 485–511

work page 2023

[7] [7]

Del Santo

D. Del Santo. Global existence and blow-up for a hyperbolic system in three space dimensions. Rend. Istit. Mat. Univ. Trieste29(1997), no. 1-2, 115–140

work page 1997

[8] [8]

Del Santo, V

D. Del Santo, V. Georgiev, E. Mitidieri. Global existence of the solutions and formation of singularities for a class of hyperbolic systems.Progr. Nonlinear Differential Equations Appl.,32 Birkhäuser Boston, Inc., Boston, MA, 1997, 117–140. 23

work page 1997

[9] [9]

Del Santo, E

D. Del Santo, E. Mitidieri. Blow-up of solutions of a hyperbolic system: the critical case.Differ. Uravn.34(1998), no. 9, 1155–1161, 1293; translation inDifferential Equations34(1998), no. 9, 1157–1163

work page 1998

[10] [10]

Dungey, A.F.M

N. Dungey, A.F.M. ter Elst, D.W. Robinson.Analysis on Lie Groups with Polynomial Growth. Progr. Math., 214 Birkäuser Boston, Inc., Boston, MA, 2003

work page 2003

[11] [11]

Fischer, M

V. Fischer, M. Ruzhansky.Quantization on Nilpotent Lie Groups. Progr. Math., 314 Birkhäuser/Springer, [Cham], 2016

work page 2016

[12] [12]

Garetto, M

C. Garetto, M. Ruzhansky. Wave equation for sums of squares on compact Lie groups.J. Differential Equations258(2015), no. 12, 4324–4347

work page 2015

[13] [13]

Georgiev, H

V. Georgiev, H. Takamura, Y. Zhou. The lifespan of solutions to nonlinear systems of a high-dimensional wave equation.Nonlinear Anal.64(2006), no. 10, 2215–2250

work page 2006

[14] [14]

R.T. Glassey. Finite-time blow-up for solutions of nonlinear wave equations.Math. Z.177 (1981), no. 3, 323–340

work page 1981

[15] [15]

Ikeda, M

M. Ikeda, M. Sobajima, K. Wakasa. Blow-up phenomena of semilinear wave equations and their weakly coupled systems.J. Differential Equations267(2019), no. 9, 5165–5201

work page 2019

[16] [16]

T. Kato. Blow-up of solutions of some nonlinear hyperbolic equations.Comm. Pure Appl. Math. 33(1980), no. 4, 501–505

work page 1980

[17] [17]

K. Kita, R. Kusaba. A remark on the blowing up of solutions to Nakao’s problem.J. Math. Anal. Appl.513(2022), no. 1, Paper No. 126199, 20 pp

work page 2022

[18] [18]

Kurokawa

Y. Kurokawa. The lifespan of radially symmetric solutions to nonlinear systems of odd dimensional wave equations.Nonlinear Anal.60(2005), no. 7, 1239–1275

work page 2005

[19] [19]

Kurokawa, H

Y. Kurokawa, H. Takamura. A weighted pointwise estimate for two dimensional wave equations and its applications to nonlinear systems.Tsukuba J. Math.27(2003), no. 2, 417–448

work page 2003

[20] [20]

Kurokawa, H

Y. Kurokawa, H. Takamura, K. Wakasa. The blow-up and lifespan of solutions to systems of semilinear wave equation with critical exponents in high dimensions.Differential Integral Equations25(2012), no. 3-4, 363–382

work page 2012

[21] [21]

Y. Li, A. Palmieri. On the blow-up of solutions to a Nakao-type problem with a time-dependent damping term. Preprint, arXiv:2510.17368

work page arXiv

[22] [22]

Y. Li, A. Palmieri. Blow-up results for a Nakao-type problem with a time-dependent damping term and derivative-type nonlinearities. Preprint, arXiv:2510.18378

work page arXiv

[23] [23]

M. Liu. Quantitative blow-up via renormalized Kato theory: resolving Nakao-type systems.J. Differential Equations462(2026), Paper No. 114165

work page 2026

[24] [24]

M. Nakao. Global existence to the initial-boundary value problem for a system of semilinear wave equations.Nonlinear Anal.146(2016), 233–257. 24

work page 2016

[25] [25]

M. Nakao. Global existence to the initial-boundary value problem for a system of nonlinear diffusion and wave equations.J. Differential Equations264(2018), no. 1, 134–162

work page 2018

[26] [26]

Narazaki

T. Narazaki. Global solutions to the Cauchy problem for the weakly coupled system of damped wave equations.Discrete Contin. Dyn. Syst.2009, suppl., Dynamical systems, differential equations and applications. 7th AIMS Conference, 592–601

work page 2009

[27] [27]

Nishihara

K. Nishihara. Asymptotic behavior of solutions for a system of semilinear heat equations and the corresponding damped wave system.Osaka J. Math.49(2012), no. 2, 331–348

work page 2012

[28] [28]

Nishihara, Y

K. Nishihara, Y. Wakasugi. Critical exponent for the Cauchy problem to the weakly coupled damped wave system.Nonlinear Anal.108(2014), 249–259

work page 2014

[29] [29]

Palmieri

A. Palmieri. On the blow-up of solutions to semilinear damped wave equations with power nonlinearity in compact Lie groups.J. Differential Equations281(2021), 85–104

work page 2021

[30] [30]

Palmieri

A. Palmieri. Semilinear wave equation on compact Lie groups.J. Pseudo-Differ. Oper. Appl. 12(2021), no. 3, Paper No. 43, 13 pp

work page 2021

[31] [31]

Palmieri

A. Palmieri. A global existence result for a semilinear wave equation with lower order terms on compact Lie groups.J. Fourier Anal. Appl.28(2022), no. 2, Paper No. 21, 15 pp

work page 2022

[32] [32]

Palmieri, H

A. Palmieri, H. Takamura. A note on blow-up results for semilinear wave equations in de Sitter and anti–de Sitter spacetimes.J. Math. Anal. Appl.514(2022), no. 1, Paper No. 126266, 40 pp

work page 2022

[33] [33]

Palmieri, H

A. Palmieri, H. Takamura. A blow-up result for a Nakao-type weakly coupled system with nonlinearities of derivative-type.Math. Ann.387(2023), no. 1-2, 111–132

work page 2023

[34] [34]

Peter, H

F. Peter, H. Weyl. Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe.Math. Ann.97(1927), no. 1, 737–755

work page 1927

[35] [35]

Ruzhansky, V

M. Ruzhansky, V. Turunen.Pseudo-differential Operators and Symmetries. Background Analysis and Advanced Topics. Pseudo Diff. Oper., 2 Birkhäuser Verlag, Basel, 2010

work page 2010

[36] [36]

Ruzhansky, N

M. Ruzhansky, N. Yessirkegenov. Hardy-Sobolev-Rellich, Hardy-Littlewood-Sobolev and Caffarelli-Kohn-Nirenberg inequalities on general Lie groups.J. Geom. Anal.34(2024), no. 7, Paper No. 223, 28 pp

work page 2024

[37] [37]

Schaeffer

J. Schaeffer. The equationutt−∆u =|u|p for the critical value ofp.Proc. Roy. Soc. Edinburgh Sect. A101(1985), no. 1-2, 31–44

work page 1985

[38] [38]

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