Blow-up result for a piezoelectric beams system with magnetic effects

Mohammad Kafini

REVIEW 4 major objections 4 minor 20 references

Blow-up result for a piezoelectric beams system with magnetic effects

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that damped piezoelectric beams with magnetic effects blow up in finite time when the initial energy is negative.

desk verdict Main blow-up theorem rests on a mis-derived inequality in Lemma 6; the paper is repairable but not publishable as written. read the letter →

arxiv 2411.13976 v1 pith:TFGPA6DY submitted 2024-11-21 math.AP

Mohammad Kafini This is my paper · ORCID

classification math.AP MSC 35B4435D3035L0535L1535L70

keywords blow-upnegativeinitialenergypiezoelectricbeamsmagneticeffectsconcavitymethodnonlinearsourcesfrictionaldampingfinite-time

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies a one-dimensional piezoelectric beam model that includes magnetic effects, two frictional dampings, and nonlinear sources, and asks whether the dampings can keep solutions bounded. It claims they cannot when the initial energy is negative and the sources dominate the dampings: the solution blows up in finite time. The proof constructs an auxiliary functional whose concavity forces it to hit zero, which makes the solution norm infinite. A separate lower-bound estimate gives a quantitative guarantee that blow-up cannot happen earlier than some computable time.

What carries the argument

The central object is the concavity functional $G(t)=F^{-\sigma}(t)$, built from $F(t)=\frac12\int_0^L(|\upsilon|^2+|p|^2)dx + \frac12 L(t)+\frac12 b(t+t_0)^2$, where $L(t)$ accumulates the damping history. The argument shows $G''(t)\leq 0$ and $G'(0)<0$, and a Taylor expansion then forces $G$ to vanish at a finite time, so $F$ becomes infinite. The inequality chain uses the energy identity $E'(t)=-\lambda_1\|\upsilon_t\|_2^2-\lambda_2\|p_t\|_2^2$ and the source-growth condition (G2) to overwhelm the damping terms.

What would settle it

Integrate system (6) numerically with a source like $I(\upsilon,p)=\frac{a}{\eta}|\upsilon-p|^{\eta}$ and initial data satisfying $E(0)<0$ and the parameter choices in the theorem; if the solution norm stays finite beyond the claimed upper bound $t_m=-G(0)/G'(0)$, the theorem's estimate is false. A direct check of Lemma 6's inequality against computed $F$, $F'$, and $E(t)$ would also identify where the argument breaks.

Watch

Extended reading notes

Core claim

Under assumptions (G1)–(G3), for initial data in the stated energy space with $E(0)<0$, the corresponding strong solution of system (6) blows up in finite time. The blow-up time is bounded above by $t_m \leq -G(0)/G'(0)$, where $G(t)=F^{-\sigma}(t)$ and $F(t)$ is the auxiliary functional in (9); the proof also bounds the blow-up time from below through an integral involving the initial value of $\int_0^L I(\upsilon,p)\,dx$ and the source growth exponents.

Load-bearing premise

The load-bearing premise is Lemma 6's bound on $I=(F'(t))^2$, used to derive $Q(t)\geq 0$, and the derivation of that bound requires the estimates $\int(|\upsilon|^2+|p|^2)dx \leq 2F(t)$ and $b^2(t+t_0)^2 \leq bF(t)$, the second of which does not follow from the definition of $F$.

Editorial extensions

If this is right

If the theorem is right, no global solution exists for this piezoelectric beam model once the initial energy is negative and conditions (G1)–(G3) hold.
The upper bound $t_m \leq -G(0)/G'(0)$ is computable from the initial data and the chosen parameters $b,t_0,\sigma$, so the result gives a concrete finite time by which blow-up must occur.
The lower-bound theorem estimates a time before which blow-up cannot happen, expressed through the initial value of the source potential $\int_0^L I(\upsilon,p)\,dx$.
The paper asserts in its conclusion that the same proof works on unbounded domains, extending the blow-up statement beyond the bounded interval.
If positive initial energy is considered with a different arrangement, the same concavity framework may still produce a blow-up result, as the conclusion suggests.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The constant $k$ in condition (20) can be read as a quantitative competition between the source exponent $\eta$ and the damping coefficients $\lambda_1,\lambda_2$; one natural next step is to identify a threshold surface separating finite-time blow-up from global existence in the full parameter space.
Because the concavity argument depends only on scalar energy inequalities, an analogous finite-dimensional system of ordinary differential equations obtained by spatial discretization should exhibit the same collapse, giving a cheap numerical test of the theorem.
The lower-bound estimate suggests a practical stopping criterion for simulations: a numerical solution that survives far beyond the computed upper bound would challenge the claimed parameter range, even if the concavity proof is accepted.
The auxiliary functional $L(t)$ includes a memory integral that is nonstandard; understanding its role may suggest other damping histories for which the blow-up result remains true with different constants.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The paper claims a finite-time blow-up result for a piezoelectric beam system with magnetic effects, frictional damping, and source terms, using Levine's concavity method. The specific system with damping on both equations hasn't been treated for blow-up before, so the result would be a legitimate new data point. The energy identity is correct, and the setup of the functional F is standard. The lower-bound section is mostly fine, modulo some notation slips.

The problem is Lemma 6, the linchpin of the proof. Its inequality (11) is derived through a string of invalid estimates. The constant lambda is never defined. The Cauchy-Schwarz step is written as an equality when it is only an inequality, and the cross term in (1/2 L' + b(t+t0))^2 is dropped without bound. More importantly, two factor-of-two errors creep in: the integral of v^2+p^2 is bounded by F, but it can only be bounded by 2F; and b^2(t+t0)^2 is bounded by bF, while the best available is 2bF. The claimed inequality (11) is stronger than anything the derivation supports; indeed for data where the integral terms in F vanish, the claimed bound fails. Since (11) is what forces Q(t) >= 0 and hence G'' <= 0, Theorem 4 collapses as written.

There is also a smaller gap: after fixing epsilon and sigma, the proof says "we may choose eta more large" to satisfy (20), but eta is fixed by assumption (G2). This is repairable by strengthening the assumptions, but as stated the theorem overclaims. The claim that blow-up holds for any negative energy without regard to the size of the damping is also not substantiated by the proof.

So the central result is not established. The paper is not nonsense; the method is appropriate, the topic is fine, and the lower-bound part is likely salvageable. But the main theorem needs a correct Lemma 6 and adjusted hypotheses. As it stands, I would not accept it, and I would not spend a referee's time on this version. If the author repairs the factor errors and states the extra assumptions, it becomes a reasonable candidate for a standard PDE journal.

Referee Report

4 major / 4 minor

Summary. The manuscript studies a one-dimensional piezoelectric beam system with magnetic effects, frictional damping, and nonlinear source terms. It claims finite-time blow-up for solutions with negative initial energy using the concavity method, and it also derives a lower bound for the blow-up time. The central argument rests on Lemma 6, which is intended to bound the square of F'(t) in terms of F(t), the kinetic energy, and the (negative) energy.

Significance. If correct, the result would extend known blow-up results to this piezoelectric model with magnetic effects and would provide both upper and lower estimates for the blow-up time. However, the proof of the key inequality in Lemma 6 is invalid, and the concavity argument collapses; the paper does not provide a correct proof of the main theorem. No numerical or machine-checkable evidence is provided.

major comments (4)

[Lemma 6, Eq. (11)] The estimate (11) is not a consequence of the preceding inequalities. From the definition of F in (9) one only has ∫(|υ|²+|p|²) ≤ 2F(t) and b²(t+t0)² = b·b(t+t0)² ≤ 2bF(t). Substituting these bounds into the displayed estimate in the proof of Lemma 6 yields I ≤ F[2(1+ε)(‖υ_t‖²+‖p_t‖²) + 4(1+1/ε)(b−λE(t))] (up to the interpretation of λ), not the factor-2-smaller expression in (11). Consequently, the claimed bound is stronger than what the proof establishes and is in general false.
[Lemma 6] The proof also contains an equality that should be an inequality: (1/2 L'(t))² = (∫∫...)(∫∫...) is not true; Cauchy-Schwarz gives ≤. Moreover, the constant λ appearing in (11) and later in (18)–(20) is never defined, so the estimate cannot be verified as written.
[Theorem 4, Eqs. (18)–(20)] Because the inequality (11) is invalid, the lower bound for Q(t) in (19) has incorrect coefficients: the multiplier of ‖υ_t‖²+‖p_t‖² should be 2(σ+1)(1+ε) rather than (σ+1)(1+ε), and the b and E(t) terms have similarly wrong factors. The choices of ε, σ, and b that lead to Q(t)≥0 are therefore based on an incorrect expression, and the conclusion G''(t)≤0 is unsupported.
[Lemma 5 and Theorem 4] The statement of Lemma 5 bounds ∫(ρυυ_tt+μpp_tt), but after the division leading to (6) the system no longer contains ρ and μ, and F'' in (14) contains ∫(υυ_tt+pp_tt) without these factors. The substitution of (10) into (15) is therefore only valid if ρ=μ=1, which is not stated; otherwise the lower bound for F'' does not follow.

minor comments (4)

[References / Theorem 1] Theorem 1 (local existence) is stated without proof and relies on a combination of results including the author's own paper [14]; this should be made precise or proved.
[Lemma 6] The text refers to 'Lemma 2.2' in the proof of Lemma 6; the lemma numbers do not match (it should be Lemma 2).
[Section 4, Eq. (25)] The constant (2/m)^r is used uniformly for all β_i; this is only an upper bound if 2/m ≥ 1, which is not guaranteed.
[Conclusion] The Conclusion contains a misspelling ('compitition') and an unsupported assertion that the same proof is valid in unbounded domains, since the proof uses Poincaré's inequality and boundary conditions.

Circularity Check

0 steps flagged · score 1.0 of 10

No substantive circularity; the concavity blow-up proof is a direct derivation, with one minor non-load-bearing self-citation for local existence.

full rationale

The blow-up argument is a direct application of the concavity method and does not reduce to its own inputs. The functional F(t) is constructed from solution norms plus an auxiliary L(t), and the proof aims to show G(t)=F^{-sigma}(t) reaches zero by deriving Q(t)=F F''-(sigma+1)(F')^2 >= 0 from the energy identity (8), the growth assumption (G2), and Young/Cauchy-Schwarz estimates. The parameters b, t0, and sigma are free parameters chosen after the hypothesis E(0)<0, so choosing them small or large to force G''<=0 and G'(0)<0 is a legitimate parameter-choice argument, not a fitted prediction. No equation is defined in terms of the blow-up conclusion, and no uniqueness theorem from the author's prior work is invoked to make the choice of method forced. The only self-citation is [14], used in Theorem 1 to support local existence; that citation is not part of the blow-up mechanism, does not forbid alternatives, and therefore is not load-bearing circularity. Reported mathematical concerns, such as the lost factor of two in Lemma 6, an undefined lambda, and the later choice of eta, concern the correctness of an estimate rather than the equivalence of premise and conclusion, so they do not constitute circularity under the stated criteria.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to data. The auxiliary constants b, t0, ε, σ in the proof are chosen by hand to close the concavity argument, not model parameters. The central claim rests on the structural assumptions (G1)-(G3) and on a local existence theorem cited from prior work including a self-citation.

assumptions (5)

domain assumption Existence of a potential I ≥ 0 with ∂I/∂υ = f1 and ∂I/∂p = f2 (assumption G1).
Invoked in Lemma 2 to identify ∫(υt f1 + pt f2) dx as d/dt∫I dx, and in Lemma 5.
domain assumption There exists η > 2 with ∫[υf1+pf2-ηI] dx ≥ 0 (G2), and the proof further needs η large enough to make (20) positive.
Used in Lemma 5 and in the proof of Theorem 4 to get Q(t) ≥ 0. The extra 'choose η more large' is not part of the theorem statement.
domain assumption Growth bounds (G3): |f1| ≤ d(|χ|^β1+|φ|^β2), |f2| ≤ d(|χ|^β3+|φ|^β4), with (n-2)βi ≤ n.
Used in the lower-bound estimate (Theorem 7).
domain assumption Theorem 1: unique strong solution exists on (0,T) for initial data in the stated space, proved by combining [6], [14], [20].
Stated without proof; [14] is a self-citation. The blow-up argument assumes a local solution exists.
standard math Bounded interval (0,L) and standard Sobolev embedding and Poincaré inequalities.
Used throughout, especially in the lower-bound section; the conclusion's claim of unbounded-domain validity conflicts with this.

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Cite this review

Pith. "Pith review of Blow-up result for a piezoelectric beams system with magnetic effects." pith.science (2026). https://pith.science/paper/TFGPA6DY

@misc{pith2026241113976,
  author       = {Pith},
  title        = {Pith review of: Blow-up result for a piezoelectric beams system with magnetic effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TFGPA6DY}},
  note         = {Machine review of arXiv:2411.13976}
}

read the original abstract

The system under studying is for a piezoelectric beams system with magnetic effects, frictional dampings and source terms. We use the concavity method to study the competition of the dampings and the sources that leads to a blow-up result for solutions with negative initial energy.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

Brunner, M

A.J. Brunner, M. Barbezat, C. Huber and P.H. Fl¨ ueler, The pot ential of active ﬁber composites made from piezoelectric ﬁbers for actuatin g and sensing applications in structural health monitoring, Materials and s truc- tures, 38 (5), 561-567, 2005. 11

work page 2005
[2]

C.Y.K. Chee, L. Tong and G.P. Steven, A review on the modelling of piezoelectric sensors and actuators incorporated in intelligent str uctures, Journal of intelligent material systems and structures, 9 (1), 3- 19, 1998

work page 1998
[3]

Destuynder, I

Ph. Destuynder, I. Legrain, L. Castel and N. Richard, Theore tical, nu- merical and experimental discussion on the use of piezoelectric dev ices for control-structure inter action, European J. Mech. A Solids 11 (2) , 181-213, 1992

work page 1992
[4]

Fern´ andez Sare, B

H.D. Fern´ andez Sare, B. Miara and M.L. Santos, A note on analyt icity to piezoelectric systems, Math. Methods Appl. Sci. 35 (18), 2157-21 65, 2012

work page 2012
[5]

Komornik, Exact Controllability and Stabilization: The Multiplier Method, Wiley, New York; 1994

V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, Wiley, New York; 1994

work page 1994
[6]

Morris and A

K. Morris and A. ¨O. ¨Ozer, Strong stabilization of piezoelectric beams with magnetic eﬀects, In 52nd IEEE Conference on Decision and Contro l, 3014- 3019, 2013

work page 2013
[7]

Ramos, C.S.L

A.J.A. Ramos, C.S.L. Gon¸ calve and S.S. Corrˆ ea Neto, Exponential stability and nu merical treatment for piezoelectric beams with magnetic eﬀe ct, ESAIM Math. Model. Numer. Anal. 52 (1), 255-274, 2018

work page 2018
[8]

Freitas, A.J.A

M.M. Freitas, A.J.A. Ramos, M.J. Dos Santos and J.L.L. Almeida, Dy- namics of piezoelectric beams with magnetic eﬀects and delay term, E vol. Equ. Control Theory 11 (2), 583-603, 2022

work page 2022

Show all 20 references

[9]

Freitas, A.J.A

M.M. Freitas, A.J.A. Ramos, A. ¨O. ¨Ozer and D.S. Almeida J´ unior, Long- time dynam ics for a fractional piezoelectric system with magnetic eﬀ ects and Fourier’s law, J. Diﬀer. Equ. 280, 891-927, 2021

2021
[10]

Fern´ andez Sare, B

H.D. Fern´ andez Sare, B. Miara and M.L. Santos, A note on analy ticity to piezoelectric systems, Math. Methods Appl. Sci. 35 (18), 2157-21 65, 2012

2012
[11]

Aﬁlal, A

M. Aﬁlal, A. Soufyane and M.L. Santos, Piezoelectric Beams with Magnetic Eﬀect and Nonlinear Damping: General Stability, DOI: 10.22541/au.170664693.37212809/v1
[12]

Ramos, A

A.J.A. Ramos, A. Ozer and M.M. Freitas, Exponential stabilization of fully dynamic and electrostatic piezoelectric beams with delayed dis- tributed damping feedback. Z. Angew. Math. Phys. 72, 26 (2021) . https://doi.org/10.1007/s00033-020-01457-8

2021 doi
[13]

Soufyane , M

A. Soufyane , M. Aﬁlal and M. L. Santos, Energy decay for a we akly non- linear damped piezoelectric beams with magnetic eﬀects and a nonlinea r delay term, Z. Angew. Math. Phys. (2021) 72:166

2021
[14]

Kaﬁni and S

M. Kaﬁni and S. Al-Omari, Local existence and lower bound of blo w-up time to a Cauchy problem of a coupled nonlinear wave equations, AIMS Mathematics, 6(8): 9059–9074. 12
[15]

Kaﬁni, S

M. Kaﬁni, S. Messaoudi, A blow-up result for a viscoelastic syste m in Rn , Electron. J. Diﬀer. Eq., 113 (2007), 1–7

2007
[16]

S. A. Messaoudi, H. Belkacem, Global nonexistence of positive in itial- energy solutions of a system of nonlinear viscoelastic wave equation s with damping and source terms, J. Math. Anal. Appl., 365 (2010), 277–2 87

2010
[17]

K. Agre, M. Rammaha, Systems of nonlinear wave equations with damping and source terms, Diﬀer. Integral Equ., 19 (2006), 1235–1270

2006
[18]

H. A. Levine, Some additional remarks on the nonexistence of g lobal solu- tions to nonlinear wave equation, SIAM J. Math. Anal., 5 (1974), 138 –146

1974
[19]

H. A. Levine, J. Serrin, A global nonexistence theorem for qua silinear evo- lution equation with dissipation, Arch. Rational Mech. Anal., 137 (199 7), 341–361
[20]

Messaoudi, A

H. Messaoudi, A. Ardjouni, S. Zitouni, H. Khochemane, Well-pos edness and general energy decay of solutions for a nonlinear damping piezo elec- tric beams system with thermal and magnetic eﬀects, Hacettepe J . Math. Stat.,Volume 52 (6) (2023), 1615 – 1630. 13

2023

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Brunner, M

A.J. Brunner, M. Barbezat, C. Huber and P.H. Fl¨ ueler, The pot ential of active ﬁber composites made from piezoelectric ﬁbers for actuatin g and sensing applications in structural health monitoring, Materials and s truc- tures, 38 (5), 561-567, 2005. 11

work page 2005

[2] [2]

C.Y.K. Chee, L. Tong and G.P. Steven, A review on the modelling of piezoelectric sensors and actuators incorporated in intelligent str uctures, Journal of intelligent material systems and structures, 9 (1), 3- 19, 1998

work page 1998

[3] [3]

Destuynder, I

Ph. Destuynder, I. Legrain, L. Castel and N. Richard, Theore tical, nu- merical and experimental discussion on the use of piezoelectric dev ices for control-structure inter action, European J. Mech. A Solids 11 (2) , 181-213, 1992

work page 1992

[4] [4]

Fern´ andez Sare, B

H.D. Fern´ andez Sare, B. Miara and M.L. Santos, A note on analyt icity to piezoelectric systems, Math. Methods Appl. Sci. 35 (18), 2157-21 65, 2012

work page 2012

[5] [5]

Komornik, Exact Controllability and Stabilization: The Multiplier Method, Wiley, New York; 1994

V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, Wiley, New York; 1994

work page 1994

[6] [6]

Morris and A

K. Morris and A. ¨O. ¨Ozer, Strong stabilization of piezoelectric beams with magnetic eﬀects, In 52nd IEEE Conference on Decision and Contro l, 3014- 3019, 2013

work page 2013

[7] [7]

Ramos, C.S.L

A.J.A. Ramos, C.S.L. Gon¸ calve and S.S. Corrˆ ea Neto, Exponential stability and nu merical treatment for piezoelectric beams with magnetic eﬀe ct, ESAIM Math. Model. Numer. Anal. 52 (1), 255-274, 2018

work page 2018

[8] [8]

Freitas, A.J.A

M.M. Freitas, A.J.A. Ramos, M.J. Dos Santos and J.L.L. Almeida, Dy- namics of piezoelectric beams with magnetic eﬀects and delay term, E vol. Equ. Control Theory 11 (2), 583-603, 2022

work page 2022