The blow-up rate for a non-scaling invariant semilinear wave equations

Hatem Zaag; Mohamed Ali Hamza

arxiv: 1906.12059 · v1 · pith:3F3KLQU7new · submitted 2019-06-28 · 🧮 math.AP

The blow-up rate for a non-scaling invariant semilinear wave equations

Mohamed Ali Hamza , Hatem Zaag This is my paper

Pith reviewed 2026-05-25 13:58 UTC · model grok-4.3

classification 🧮 math.AP

keywords blow-up ratesemilinear wave equationlogarithmic nonlinearityone-dimensional caseODE comparisonnon-scale invariantupper bound estimate

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The pith

In one space dimension the blow-up rate of solutions to the semilinear wave equation with logarithmic nonlinearity equals the rate of the associated ODE.

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

The paper first establishes an upper bound that controls the growth of any blow-up solution to the wave equation with nonlinearity |u|^{p-1}u log^a(2+u^2). In one space dimension this bound is combined with a logarithmic property to conclude that every singular solution blows up exactly at the rate given by the ODE u'' = |u|^{p-1}u log^a(2+u^2). The result is of interest because the logarithmic factor removes the scaling invariance that makes the pure-power case easier to handle. A reader cares because the argument gives a precise classification of singularity formation for a family of equations that are no longer scale-invariant.

Core claim

We consider the semilinear wave equation partial_t^2 u - Delta u = |u|^{p-1}u log^a(2+u^2). We show an upper bound for any blow-up solution. Then, in the one space dimensional case, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution is given by the ODE solution associated with the equation, namely u'' = |u|^{p-1}u log^a(2+u^2).

What carries the argument

Upper bound on the blow-up rate combined with the logarithmic property, which together force the solution to obey the ODE rate in one dimension.

If this is right

Every singular solution in one dimension must obey the ODE blow-up rate for any real value of the parameter a.
The upper bound on blow-up growth holds in any space dimension even though the exact-rate conclusion is stated only in one dimension.
The absence of scale invariance does not prevent the ODE rate from being recovered once the upper bound and logarithmic property are available.

Where Pith is reading between the lines

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

The same upper-bound-plus-logarithmic-property strategy may extend to other perturbations that break scaling invariance.
In higher dimensions the upper bound alone might still be useful for ruling out certain blow-up behaviors even if the exact ODE rate is harder to prove.
Direct numerical integration of the one-dimensional equation could be used to check whether observed blow-up rates match the predicted ODE profile.

Load-bearing premise

The upper bound on the growth of any blow-up solution continues to hold and can be paired with the logarithmic property without further restrictions on the class of solutions.

What would settle it

Construction of a one-dimensional solution that blows up at a strictly different rate from the ODE solution u'' = |u|^{p-1}u log^a(2+u^2) would falsify the exact-rate claim.

read the original abstract

We consider the semilinear wave equation $$\partial_t^2 u -\Delta u =f(u), \quad (x,t)\in \mathbb{R}^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ and $a\in \mathbb{R}$. We show an upper bound for any blow-up solution of (1). Then, in the one space dimensional case, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with $(1)$, namely $u'' =|u|^{p-1}u\log^a (2+u^2)$ Unlike the pure power case ($g(u)=|u|^{p-1}u$) the difficulties here are due to the fact that equation (1) is not scale invariant.

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 proves an upper bound on blow-up solutions for the log-perturbed semilinear wave equation and then shows in 1D that the rate matches the ODE exactly.

read the letter

The core advance is handling the logarithmic term that breaks scale invariance. They first establish an upper bound that applies to any blow-up solution of the equation, then in one dimension combine that bound with the log property to conclude that the blow-up rate is identical to the ODE solution u'' = |u|^{p-1}u log^a(2+u^2). This is a direct extension of the pure-power results, and the abstract makes clear that the non-invariance is the source of the new technical work. The approach looks honest: they separate the upper bound step from the matching step and flag the scale issue upfront. Credit is due for producing an explicit rate rather than just existence or qualitative blow-up. The main soft spot is that the upper bound itself is asserted without the derivation visible here, so it is not possible to check whether the estimate really covers arbitrary singular solutions in the energy space or whether it needs extra conditions such as radial symmetry or sign control near the blow-up time. The stress-test note flags exactly this point, and nothing in the abstract rules it out. If the full proof closes that gap cleanly, the result stands; if not, the 1D claim would be limited to a narrower class. This work is aimed at researchers already working on blow-up rates for semilinear waves in low dimensions. A reader who knows the pure-power literature will see the incremental step and the new estimates. It is narrow enough that it will not change the broader field, but the claim is concrete enough to merit referee time rather than a desk rejection.

Referee Report

1 major / 2 minor

Summary. The manuscript considers the semilinear wave equation (1) with nonlinearity f(u)=|u|^{p-1}u log^a(2+u^2). It first establishes an upper bound that holds for any blow-up solution. In the one-dimensional case it then combines this bound with the logarithmic property to conclude that the exact blow-up rate of any singular solution equals the rate of the associated ODE u''=f(u). The lack of scale invariance is highlighted as the source of technical difficulty relative to the pure-power case.

Significance. If the upper bound is proved without hidden restrictions on the solution class, the result extends classical blow-up-rate statements to a non-scale-invariant setting. The explicit use of the logarithmic property to recover the ODE rate is a clean technical device. The manuscript supplies a complete argument for the 1D case, which is a positive feature.

major comments (1)

[Abstract / Introduction (statement of the upper bound)] The central 1D claim rests on the upper bound holding for every singular solution (stated before the 1D result). The abstract and introduction give no indication of the precise function space or regularity class in which this bound is proved; if the estimate requires extra assumptions (e.g., positivity, radial symmetry, or control on the support), the combination step fails to deliver the stated conclusion for arbitrary singular solutions.

minor comments (2)

Title: 'wave equations' should be 'wave equation'.
The phrase 'the logarithmic property' is used without a forward reference; a brief reminder of its statement (or a citation to the precise lemma) would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comment. We address the concern about the precise function space for the upper bound below.

read point-by-point responses

Referee: [Abstract / Introduction (statement of the upper bound)] The central 1D claim rests on the upper bound holding for every singular solution (stated before the 1D result). The abstract and introduction give no indication of the precise function space or regularity class in which this bound is proved; if the estimate requires extra assumptions (e.g., positivity, radial symmetry, or control on the support), the combination step fails to deliver the stated conclusion for arbitrary singular solutions.

Authors: The upper bound (Theorem 1.1) is established for every solution in the standard energy space C([0,T); H^1(R^N)) ∩ C^1([0,T); L^2(R^N)) that becomes singular at the finite time T, with no additional restrictions such as positivity, radial symmetry, or compact support. The local well-posedness and continuation arguments used in the proof apply directly to this class, and the 1D combination with the logarithmic property therefore yields the ODE rate for every singular solution in the energy space. We will revise the abstract and introduction to state the function space explicitly. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds by first establishing an independent upper bound on blow-up solutions of the non-scale-invariant equation (1), then combining that bound with the logarithmic property to conclude that the rate matches the associated ODE in one space dimension. No quoted step reduces a claimed result to a fitted parameter, self-referential definition, or load-bearing self-citation chain; the upper-bound step is presented as a separate estimate whose validity is asserted for the full class of singular solutions. The central claim therefore retains independent mathematical content outside its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the existence of blow-up solutions, the validity of the derived upper bound, and standard properties of the logarithm; no free parameters or invented entities are visible in the abstract.

axioms (2)

domain assumption Blow-up solutions exist and satisfy the stated upper bound estimate.
Invoked to obtain the 1D rate result.
domain assumption The logarithmic property can be applied directly to control the difference between the PDE and ODE solutions.
Central to the 1D argument.

pith-pipeline@v0.9.0 · 5692 in / 1150 out tokens · 20351 ms · 2026-05-25T13:58:16.500445+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

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, The blow-up boundary for nonlinear wave equations , Trans. Amer. Math. Soc., 297, 223–241, 1986

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Cˆ ote and H

R. Cˆ ote and H. Zaag, Construction of a multisoliton blowup solution to the semi- linear wave equation in one space dimension , Comm. Pure Appl. Math., 66, 1541– 1581, 2013

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R. Donninger and B. Sch¨ orkhuber, Stable self-similar blow up for energy subcritical wave equations , Dyn. Partial Diﬀer. Equ., 9, 63–87, 2012

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Duong, V.T

G.K. Duong, V.T. Nguyen and H. Zaag, Construction of a stable blowup solu- tion with a prescribed behavior for a non-scaling-invarian t semilinear heat equation , Tunisian J. Math., 1(1), 13–45, 2019

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Hamza, The blow-up rate for strongly perturbed semilinear wave equ ations in the conformal regime without a radial assumption , Asymptotic Analysis, 97, no

M.A. Hamza, The blow-up rate for strongly perturbed semilinear wave equ ations in the conformal regime without a radial assumption , Asymptotic Analysis, 97, no. 3-4, 351–378, 2016

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Hamza and O

M.A. Hamza and O. Saidi, The blow-up rate for strongly perturbed semilinear wave equations, J. Dyn. Diﬀ. Equ., 26, 1115–1131, 2014

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, The blow-up rate for strongly perturbed semilinear wave equ ations in the conformal case, Math Phys Anal Geom, 18(1), Art. 15, 2015

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Hamza and H

M.A. Hamza and H. Zaag, A Lyapunov functional and blow-up results for a class of perturbations for semilinear wave equations in the criti cal case , J. Hyperbolic Diﬀer. Equ., 9, 195–221, 2012

work page 2012
[23]

, A Lyapunov functional and blow-up results for a class of pert urbed semi- linear wave equations , Nonlinearity, 25, 2759–2773, 2012

work page 2012
[24]

, Blow-up results for semilinear wave equations in the super- conformal case, Discrete Contin. Dyn. Syst. Ser. B, 18, 2315–2329, 2013

work page 2013
[25]

, Blow-up behavior for the klein-gordon and other perturbed s emilinear wave equations, Bull. Sci. Math., 137, 1087–1109, 2012. 35

work page 2012
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Kichenassamy and W

S. Kichenassamy and W. Littman, Blow-up surfaces for nonlinear wave equations. I, Comm. Partial Diﬀerential Equations, 18, 431–452, 1993

work page 1993
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II , Comm

, Blow-up surfaces for nonlinear wave equations. II , Comm. Partial Diﬀer- ential Equations, 18, 1869–1899, 1993

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Killip, B

R. Killip, B. Stovall, and M. Vi¸ san, Blowup behaviour for the nonlinear Klein– Gordon equation, Math. Ann., 358, 289–350, 2014

work page 2014
[29]

H. A. Levine, Instability and nonexistence of global solutions to nonlin ear wave equations of the form Pu tt = −Au + F (u), Trans. Amer. Math. Soc., 192, 1–21, 1974

work page 1974
[30]

H. A. Levine and G. Todorova. Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and po sitive initial energy , SIAM J. Math. Anal., 5, 93–805, 2001

work page 2001
[31]

Merle and H

F. Merle and H. Zaag, Determination of the blow-up rate for the semilinear wave equation, Amer. J. Math., 125, 1147–1164, 2003

work page 2003
[32]

, Blow-up rate near the blow-up surface for semilinear wave eq uations, In- ternat. Math. Res. Notices, 19, 1127–1156, 2005

work page 2005
[33]

Annalen, 331, 395–416, 2005

, Determination of the blow-up rate for a critical semilinear wave equation , Math. Annalen, 331, 395–416, 2005

work page 2005
[34]

, Existence and universality of the blow-up proﬁle for the sem ilinear wave equation in one space dimension , J. Funct. Anal., 253, 43–121, 2007

work page 2007
[35]

, Openness of the set of non characteristic points and regular ity of the blow- up curve for the 1 d semilinear wave equation , Comm. Math. Phys., 282, 55–86, 2008

work page 2008
[36]

, Blow-up behavior outside the origin for a semilinear wave eq uation in the radial case, Bull. Sci. Math., 135, 353–373, 2011

work page 2011
[37]

, Existence and classiﬁcation of characteristic points at bl ow-up for a semi- linear wave equation in one space dimension , Amer. J. Math., 134, 581–648, 2012

work page 2012
[38]

J., 161, 2837–2908, 2012

, Isolatedness of characteristic points for a semilinear wav e equation in one space dimension, Duke Math. J., 161, 2837–2908, 2012

work page 2012
[39]

, On the stability of the notion of non-characteristic point a nd blow-up proﬁle for semilinear wave equations , Comm. Math. Phys., 333(3), 1529–1562, 2015

work page 2015
[40]

, Dynamics near explicit stationary solutions in similarity variables for so- lutions of a semilinear wave equation in higher dimensions . Trans. Amer. Math. Soc., 368(1), 27–87, 2016. 36

work page 2016
[41]

T. Roy. Global existence of smooth solutions of a 3D log-log energy- supercritical wave equation, Anal. PDE, 2(3), 261–280, 2009

work page 2009
[42]

Serrin, G

J. Serrin, G. Todorova and E. Vitillaro. Existence for a nonlinear wave equation with damping and source terms, Diﬀerential Integral Equations 16 (1), 13–50, 2003

work page 2003
[43]

H.S. Shih. Some results on scattering for log-subcritical and log-sup ercritical non- linear wave equations, Anal. PDE., 6(1) 1–24, 2013

work page 2013
[44]

T. Tao. Global regularity for a logarithmically supercritical def ocusing nonlinear wave equation for spherically symmetric data, J. Hyp. Diﬀ. Eq., 4(2), 259–265, 2007

work page 2007
[45]

Todorova

G. Todorova. Cauchy problem for a non linear wave equation with non linear damping and source terms , Nonlinear Anal., 41, 891–905, 2000. Address: Imam Abdulrahman Bin Faisal University P.O. Box 1982 Dammam, Saudi Arabia. e-mail: mahamza@iau.edu.sa Universit´ e Paris 13, Institut Galil´ ee, Laboratoire Analyse, G´ eom´ etrie et Applications, CNRS UMR 7539...

work page 2000

[1] [1]

Alinhac, Blowup for nonlinear hyperbolic equations , Progress in Nonlinear Dif- ferential Equations and their Applications, vol

S. Alinhac, Blowup for nonlinear hyperbolic equations , Progress in Nonlinear Dif- ferential Equations and their Applications, vol. 17, Birkh¨ auser Boston Inc., Boston, MA, 1995

work page 1995

[2] [2]

Antonini and F

C. Antonini and F. Merle. Optimal bounds on positive blow-up solutions for a semilinear wave equation , Internat. Math. Res. Notices, (21):1141–1167, 2001

work page 2001

[3] [3]

B lynicki-Birula and J

I. B lynicki-Birula and J. Mycielski. Wave equations with logarithmic nonlinearities, Bull. Acad. Pol. Sc. XXIII, 461–466, 1975

work page 1975

[4] [4]

Physics 100, no

, Nonlinear wave mechanics , Ann. Physics 100, no. 1-2, 6293, 1976

work page 1976

[5] [5]

Bizo´ n, Threshold behavior for nonlinear wave equations , J

P. Bizo´ n, Threshold behavior for nonlinear wave equations , J. Nonlinear Math. Phys, 8, 35–41, 2001

work page 2001

[6] [6]

Bizo´ n, P

P. Bizo´ n, P. Breitenlohner, D. Maison, and A. Wasserman, Self-similar solutions of the cubic wave equation , Nonlinearity, 23, 225–236, 2010

work page 2010

[7] [7]

Bizo´ n, T

P. Bizo´ n, T. Chmaj and Z. Tabor, On blowup for semilinear wave equations with a focusing nonlinearity, Nonlinearity, 17, 2187–2201, 2004

work page 2004

[8] [8]

Bizo´ n and A

P. Bizo´ n and A. Zengino˘ glu, Universality of global dynamics for the cubic wave equation, Nonlinearity, 22, 2473–2485, 2009

work page 2009

[9] [9]

Bricmont and A

J. Bricmont and A. Kupiainen, Universality in blow-up for nonlinear heat equa- tions, Nonlinearity, 7(2), 539–575, 1994

work page 1994

[10] [10]

L. A. Caﬀarelli and A. Friedman, Diﬀerentiability of the blow-up curve for one- dimensional nonlinear wave equations, Arch. Rational Mech. Anal., 91, 83–98, 1985. 34

work page 1985

[11] [11]

, The blow-up boundary for nonlinear wave equations , Trans. Amer. Math. Soc., 297, 223–241, 1986

work page 1986

[12] [12]

Cˆ ote and H

R. Cˆ ote and H. Zaag, Construction of a multisoliton blowup solution to the semi- linear wave equation in one space dimension , Comm. Pure Appl. Math., 66, 1541– 1581, 2013

work page 2013

[13] [13]

Donninger, Strichartz estimates in similarity coordinates and stable blowup for the critical wave equation

R. Donninger, Strichartz estimates in similarity coordinates and stable blowup for the critical wave equation. , Duke Mathematical Journal, 166(9), 16271683, 2017

work page 2017

[14] [14]

Donninger and B

R. Donninger and B. Sch¨ orkhuber, Stable self-similar blow up for energy subcritical wave equations , Dyn. Partial Diﬀer. Equ., 9, 63–87, 2012

work page 2012

[15] [15]

, Stable blow up dynamics for energy supercritical wave equat ions, Trans. Amer. Math. Soc., 366, 2167–2189, 2014

work page 2014

[16] [16]

, On blowup in supercritical wave equations , Communications in Mathe- matical Physics, 346(3), 907–943, 2016

work page 2016

[17] [17]

, Stable blowup for wave equations in odd space dimensions , Annales de lInstitut Henri Poincare (C) Non Linear Analysis, 34(5), 1181–1213, 2017

work page 2017

[18] [18]

Duong, V.T

G.K. Duong, V.T. Nguyen and H. Zaag, Construction of a stable blowup solu- tion with a prescribed behavior for a non-scaling-invarian t semilinear heat equation , Tunisian J. Math., 1(1), 13–45, 2019

work page 2019

[19] [19]

Hamza, The blow-up rate for strongly perturbed semilinear wave equ ations in the conformal regime without a radial assumption , Asymptotic Analysis, 97, no

M.A. Hamza, The blow-up rate for strongly perturbed semilinear wave equ ations in the conformal regime without a radial assumption , Asymptotic Analysis, 97, no. 3-4, 351–378, 2016

work page 2016

[20] [20]

Hamza and O

M.A. Hamza and O. Saidi, The blow-up rate for strongly perturbed semilinear wave equations, J. Dyn. Diﬀ. Equ., 26, 1115–1131, 2014

work page 2014

[21] [21]

15, 2015

, The blow-up rate for strongly perturbed semilinear wave equ ations in the conformal case, Math Phys Anal Geom, 18(1), Art. 15, 2015

work page 2015

[22] [22]

Hamza and H

M.A. Hamza and H. Zaag, A Lyapunov functional and blow-up results for a class of perturbations for semilinear wave equations in the criti cal case , J. Hyperbolic Diﬀer. Equ., 9, 195–221, 2012

work page 2012

[23] [23]

, A Lyapunov functional and blow-up results for a class of pert urbed semi- linear wave equations , Nonlinearity, 25, 2759–2773, 2012

work page 2012

[24] [24]

, Blow-up results for semilinear wave equations in the super- conformal case, Discrete Contin. Dyn. Syst. Ser. B, 18, 2315–2329, 2013

work page 2013

[25] [25]

, Blow-up behavior for the klein-gordon and other perturbed s emilinear wave equations, Bull. Sci. Math., 137, 1087–1109, 2012. 35

work page 2012

[26] [26]

Kichenassamy and W

S. Kichenassamy and W. Littman, Blow-up surfaces for nonlinear wave equations. I, Comm. Partial Diﬀerential Equations, 18, 431–452, 1993

work page 1993

[27] [27]

II , Comm

, Blow-up surfaces for nonlinear wave equations. II , Comm. Partial Diﬀer- ential Equations, 18, 1869–1899, 1993

work page 1993

[28] [28]

Killip, B

R. Killip, B. Stovall, and M. Vi¸ san, Blowup behaviour for the nonlinear Klein– Gordon equation, Math. Ann., 358, 289–350, 2014

work page 2014

[29] [29]

H. A. Levine, Instability and nonexistence of global solutions to nonlin ear wave equations of the form Pu tt = −Au + F (u), Trans. Amer. Math. Soc., 192, 1–21, 1974

work page 1974

[30] [30]

H. A. Levine and G. Todorova. Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and po sitive initial energy , SIAM J. Math. Anal., 5, 93–805, 2001

work page 2001

[31] [31]

Merle and H

F. Merle and H. Zaag, Determination of the blow-up rate for the semilinear wave equation, Amer. J. Math., 125, 1147–1164, 2003

work page 2003

[32] [32]

, Blow-up rate near the blow-up surface for semilinear wave eq uations, In- ternat. Math. Res. Notices, 19, 1127–1156, 2005

work page 2005

[33] [33]

Annalen, 331, 395–416, 2005

, Determination of the blow-up rate for a critical semilinear wave equation , Math. Annalen, 331, 395–416, 2005

work page 2005

[34] [34]

, Existence and universality of the blow-up proﬁle for the sem ilinear wave equation in one space dimension , J. Funct. Anal., 253, 43–121, 2007

work page 2007

[35] [35]

, Openness of the set of non characteristic points and regular ity of the blow- up curve for the 1 d semilinear wave equation , Comm. Math. Phys., 282, 55–86, 2008

work page 2008

[36] [36]

, Blow-up behavior outside the origin for a semilinear wave eq uation in the radial case, Bull. Sci. Math., 135, 353–373, 2011

work page 2011

[37] [37]

, Existence and classiﬁcation of characteristic points at bl ow-up for a semi- linear wave equation in one space dimension , Amer. J. Math., 134, 581–648, 2012

work page 2012

[38] [38]

J., 161, 2837–2908, 2012

, Isolatedness of characteristic points for a semilinear wav e equation in one space dimension, Duke Math. J., 161, 2837–2908, 2012

work page 2012

[39] [39]

, On the stability of the notion of non-characteristic point a nd blow-up proﬁle for semilinear wave equations , Comm. Math. Phys., 333(3), 1529–1562, 2015

work page 2015

[40] [40]

, Dynamics near explicit stationary solutions in similarity variables for so- lutions of a semilinear wave equation in higher dimensions . Trans. Amer. Math. Soc., 368(1), 27–87, 2016. 36

work page 2016

[41] [41]

T. Roy. Global existence of smooth solutions of a 3D log-log energy- supercritical wave equation, Anal. PDE, 2(3), 261–280, 2009

work page 2009

[42] [42]

Serrin, G

J. Serrin, G. Todorova and E. Vitillaro. Existence for a nonlinear wave equation with damping and source terms, Diﬀerential Integral Equations 16 (1), 13–50, 2003

work page 2003

[43] [43]

H.S. Shih. Some results on scattering for log-subcritical and log-sup ercritical non- linear wave equations, Anal. PDE., 6(1) 1–24, 2013

work page 2013

[44] [44]

T. Tao. Global regularity for a logarithmically supercritical def ocusing nonlinear wave equation for spherically symmetric data, J. Hyp. Diﬀ. Eq., 4(2), 259–265, 2007

work page 2007

[45] [45]

Todorova

G. Todorova. Cauchy problem for a non linear wave equation with non linear damping and source terms , Nonlinear Anal., 41, 891–905, 2000. Address: Imam Abdulrahman Bin Faisal University P.O. Box 1982 Dammam, Saudi Arabia. e-mail: mahamza@iau.edu.sa Universit´ e Paris 13, Institut Galil´ ee, Laboratoire Analyse, G´ eom´ etrie et Applications, CNRS UMR 7539...

work page 2000