On the wave equation with variable exponent nonlinearity and distributive delay

Mohammad Kafini

REVIEW 4 major objections 4 minor 28 references

On the wave equation with variable exponent nonlinearity and distributive delay

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

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Pith's one-line read This paper proves that a wave equation with variable-exponent nonlinearity and distributed delay blows up in finite time when initial energy is negative, and otherwise decays at explicit polynomial or exponential rates.

desk verdict A plausible, likely correctable program for variable-exponent wave equations with distributed delay, but missing well-posedness and wrong-looking inequality displays make it conditional, not acceptable as written. read the letter →

arxiv 2411.15528 v1 pith:LIO7AXMY submitted 2024-11-23 math.AP

Mohammad Kafini This is my paper · ORCID

classification math.AP MSC 35B0535L0535L1535L70

keywords variableexponentwaveequationdistributeddelaynonlineardampingblow-upglobalexistenceenergydecaylifespan

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 nonlinear wave equation whose damping combines an instantaneous term and a distributed-delay term, both with a variable exponent, alongside a variable-exponent source, under the condition that the total delay weight stays below the instantaneous damping coefficient. It proves that when the initial energy $E(0)$ is negative, the solution cannot exist for all time, and the nonexistence time is bounded above by an explicit quantity built from the initial data and the exponent bounds. It also proves that sufficiently small initial data give a global solution, and that every global solution decays at the explicit rate $E(t)\le cE(0)/(1+t)^{2/(m_2-2)}$ when $m_2>2$, and exponentially when $m(\cdot)=2$. The significance is quantitative: the delay does not merely preserve the known blow-up/decay picture, it leaves that picture with explicit numerical bounds a reader can test.

What carries the argument

The load-bearing object is the memory variable $z(x,\rho,t,\tau)=u_t(x,t-\tau\rho)$, which transforms the distributed delay into a transport equation $\tau z_t+z_\rho=0$ over the delay interval $(\tau_1,\tau_2)$ and the history variable $\rho\in(0,1)$. The energy functional $E(t)$ includes a weighted integral of $z$ with weight $\tau(\mu_2(\tau)+\xi(x))/m(x)$, and the dissipativity condition $\int_{\tau_1}^{\tau_2}\mu_2(\tau)\,d\tau+(\tau_2-\tau_1)\xi(x)/m(x)<\mu_1$ makes $E$ nonincreasing. For blow-up, the proof differentiates $L(t)=H^{1-\alpha}(t)+\varepsilon\int_\Omega uu_t\,dx$ with $H=-E$ and uses a weighted convexity inequality with a time-dependent parameter to absorb the delay terms, yielding $L'(t)\ge\chi L^{1/(1-\alpha)}(t)$. For decay, multiplying the equation by $uE^q(t)$ produces the integral inequality $\int_s^\infty E^{q+1}(t)\,dt\le cE(s)$, which a standard decay lemma converts into the stated polynomial or exponential rates.

What would settle it

A numerical experiment in a bounded domain with $n\ge3$, exponents satisfying (2), delay weight satisfying (4), and $E(0)<0$, that observes a solution surviving beyond $(1-\alpha)/(\chi\alpha)[L(0)]^{\alpha/(1-\alpha)}$ would contradict Theorem 15; a small-data solution whose energy decays slower than $cE(0)/(1+t)^{2/(m_2-2)}$ for $m_2>2$ would contradict Theorem 21.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a wave equation with variable-exponent source and damping, plus a distributed delay in the damping term, admits the same pair of opposing phenomena as the undelayed problem: nonexistence (blow-up) when the initial energy is negative, with an explicit upper bound on the lifespan, and global existence with uniform decay when the initial data satisfy a smallness condition. The decay theorem is sharp in form: the rate is polynomial, $E(t)\le cE(0)/(1+t)^{2/(m_2-2)}$, when the upper exponent $m_2$ exceeds $2$, and exponential when $m(\cdot)=2$. The proofs run through a modified energy functional, a memory variable that converts the distributed delay into a transport equation, and a multiplier functional whose differential inequality forces either finite-time nonexistence or the integral decay estimate needed for the rates.

Load-bearing premise

The proofs assume strong solutions in the class of Definition 7 exist on the relevant time interval, citing a similar problem rather than proving well-posedness for this equation; if such solutions do not exist, the energy identities and all derived bounds have nothing to attach to.

Editorial extensions

If this is right

Under $E(0)<0$, no solution can continue past the explicit time $(1-\alpha)/(\chi\alpha)[L(0)]^{\alpha/(1-\alpha)}$, because the differential inequality $L'(t)\ge\chi L^{1/(1-\alpha)}(t)$ forces a singularity in finite time.
Under the small-data condition (31), the global solution stays uniformly bounded in the energy norm, so the model does not blow up for small initial energy.
The decay theorem gives a convergence rate that is polynomial when the damping exponent can exceed $2$ and exponential when it is identically $2$, matching the constant-exponent intuition.
The lower lifespan bound of Theorem 16 provides an integral expression depending only on the initial $L^{p(\cdot)}$-weight and the energy, which can be used as a worst-case estimate for the maximal existence time.
All results require the delay-dominance condition (10); if it fails, the energy need not be dissipative and the statements may cease to hold.

Reading between the lines

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

Because the paper's bounds are explicit in the exponent bounds and delay weight, they give a direct way to compare the theory with simulations: the lifespan bound can be checked independently of the analytic construction.
The condition (4)/(10) is likely close to sharp: for pointwise delays in the constant-exponent case, delay weights at or above the damping coefficient are known to produce instability, so the same threshold should be expected for the distributed, variable-exponent setting.
The multiplier construction may extend to other damped evolution equations with variable-exponent nonlinearity and distributed delay, such as viscoelastic or thermoelastic systems, giving analogous lifespan and decay bounds.

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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

Short version: the paper is a recognizable, likely correctable program, but it is not ready as written. The combination of distributed delay with variable-exponent damping and source is genuinely new, and the paper gives a complete qualitative picture: blow-up with an explicit lifespan bound, global existence for small initial data, and polynomial/exponential decay. The proofs follow the Messaoudi–Talahmeh and Nicaise–Pignotti template, but the variable-exponent adjustments are real and the energy functional is coherent.

The main problems are two, and they are both load-bearing. First, Definition 7 defines a strong-solution class and sends the reader to [11] for well-posedness. [11] treats a linear constant-exponent distributed-delay equation; it does not establish existence, uniqueness, or regularity for this variable-exponent problem. Since Theorems 15, 18, and 21 all start with 'let u be a solution of (8),' the theorems currently have no proven object to attach to. That is an omission, not an internal contradiction, but it is the first thing a referee should demand. Second, some displayed inequalities are wrong as written. Lemma 8 sets C0 to a maximum of two infima, while the conclusion E' ≤ −C0[...] needs the minimum; a maximum can fail. Inequality (25) shows k^{m1−1} in a coefficient that, from the preceding estimate (23), should carry k^{1−m1}; with the stated 'large k' choice, the displayed sign would make the coefficient negative, not positive. These look like fixable slips rather than fatal flaws, but they block certification.

I agree with the reader's conditional verdict. The proof skeleton is standard, the technical lemmas are imported rather than proved, and the missing well-posedness is the biggest gap. It is not a paper to accept as is. It is also not a paper to desk reject: the core ideas are sound, the novelty is real for the subfield, and a serious referee could work through the corrections. If I worked on variable-exponent hyperbolic PDEs, I would want to see a revised version; I would not cite this version yet.

Referee Report

4 major / 4 minor

Summary. The paper considers a wave equation on a bounded domain with a variable-exponent nonlinear damping term, a distributed delay in the damping, and a variable-exponent source term. The author defines an energy functional, proves a dissipation inequality under a smallness condition on the delay weight, and then derives three main results: finite-time blow-up for negative initial energy with an explicit upper bound on the nonexistence time, a lower bound for the lifespan, and global existence with polynomial or exponential decay for global solutions. The proofs follow standard multiplier-and-energy methods, with several technical lemmas quoted from earlier works by the author and by Messaoudi and collaborators.

Significance. If fully established, the results would extend known constant-exponent blow-up and decay theorems for wave equations with delay to the variable-exponent setting with distributed delay, which appears to be a new combination. The paper is clearly written in structure and uses the standard toolbox of variable-exponent Lebesgue/Sobolev spaces. However, the manuscript currently has several load-bearing gaps: the solution class in Definition 7 is not shown to exist for the actual problem, the dissipation constant in Lemma 8 is defined with the wrong extremum, the key displayed inequality (25) in the blow-up proof contains incorrect exponents and constants, and Theorem 21 states decay for arbitrary global solutions while its proof relies on the smallness condition (31) that is not assumed in the theorem. These issues prevent the central claims from being accepted in their present form, although most appear repairable.

major comments (4)

[Section 3, Definition 7] The paper defines strong solutions of (8) but does not prove local well-posedness for this specific problem. The sentence "See [11] for the well-posedness of a similar problems" is not sufficient: reference [11] concerns a linear wave equation with constant coefficients and constant delay, without the variable-exponent nonlinear damping, the distributed-delay term with the same variable exponent, or the superlinear source. No compatibility conditions on the history data f0 are given. Since Theorems 15, 18, and 21 all start from "let u be a solution of (8)", the central claims are conditional on an unproved existence and regularity result. In particular, Theorem 18 asserts that the solution is global in time, but without local existence there is no object to continue globally.
[Section 3, Lemma 8, Eq. (12)] The constant C0 is defined as C0 = max{ inf f(x), inf ξ(x)/m(x) }. To obtain E'(t) ≤ -C0[∫|ut|^m + ∫∫|z(1)|^m] from the preceding estimate E'(t) ≤ -∫ f(x)|ut|^m - ∫∫ ξ(x)/m(x)|z(1)|^m, the constant must be the minimum of the two infima, not the maximum. With the proposed definition the displayed inequality can fail when one of the two terms is the smaller one. This error propagates into the dissipation used in the blow-up proof and in the decay proof (e.g., Eq. (46)). The fix is straightforward (replace max by min), but as written the lemma is incorrect.
[Section 4, Eq. (25)] Displayed inequality (25) contains several incorrect factors. From (23) the factor in the u|u|^m terms is k^{1-m1}, not k^{m1-1}; additionally the coefficient involving the delay should be proportional to µ1 (or to µ1 times (τ2-τ1), depending on how the Young bound is written), not C/[m1 µ1(τ2-τ1)] k^{m1-1}. As a consequence, the sentence "The constant k is chosen to be large" is inconsistent with the displayed expression, since a large k would make the negative term involving k^{m1-1} large and would destroy the positivity needed for (26). The argument can likely be repaired by using the correct exponent k^{1-m1}, but the proof as written does not establish the key inequality that yields (26).
[Section 5, Theorem 21] Theorem 21 is stated for any global solution under only conditions (2) and (3), but its proof uses the smallness condition (31) from Lemma 17 to control the source term at Eq. (43), namely ∫|u|^{p(x)} dx ≤ β ||∇u||_2^2. Without assuming I(0)>0 and β < (p1-2)/(2p1), the last two terms in (36) cannot be absorbed into the left-hand side, and the derivation of (44)-(46) is invalid. The theorem statement must either include the hypotheses of Lemma 17 (or of Theorem 18) or the proof must provide a different argument for arbitrary global solutions. As stated, the theorem is unsupported.

minor comments (4)

[Section 2, Eq. (20)-(21)] In inequality (21), the factor µ1(τ2-τ1) is not justified by the assumption ∫ τ1^τ2 µ2(τ)dτ < µ1; the sum of µ2 over the interval is bounded by µ1, not by µ1(τ2-τ1) unless additional boundedness of µ2 is assumed. This should be corrected or clarified, and the ensuing constants in (25) adjusted accordingly.
[Section 4, Theorem 16] The integral in Theorem 16 has denominator y + E(0) + c y^{p2-1} + c y^{p1-1}. If E(0) < 0, as in the blow-up result of Theorem 15, the denominator can be negative or zero for small y, making the integral improper. The theorem should specify the range of integration where the denominator is positive, or assume E(0) ≥ 0.
[Throughout] The manuscript contains numerous typographical errors, including "Consequantely", "distibutive", "Existance" in the Section 5 heading, "strat" in the text, and inconsistent use of "lifespan". A careful proofreading pass is needed.
[Section 5, Eq. (39)] In the estimate of the third term in (36), the intermediate expression contains c(ε)∫(-E')^{2(q+1)/m2} dt; this equals c(ε)∫(-E') dt only after using the choice q = m2/2 - 1, which yields 2(q+1)/m2 = 1. The text should state this explicitly, as otherwise the reader cannot see how the term is bounded by c(ε)E(s).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the blow-up, global-existence, and decay proofs are self-contained energy arguments; self-citations are technical only and not load-bearing.

full rationale

The derivation chain is not circular. Theorem 15 constructs L(t)=H^{1-α}(t)+ε∫Ω uu_t dx from H(t)=-E(t), and pure energy, Young, Hölder, and embedding estimates yield L'(t) ≥ χ L^{1/(1-α)}(t); finite-time nonexistence follows by direct integration. The bound (1-α)/(χ α)[L(0)]^{α/(1-α)} is an output of that differential inequality, not an input. Theorem 18 with Lemma 17 uses a standard invariant-set argument with I(t), J(t), and E(t); no parameter is fitted to force the result. Theorem 21 derives ∫_s^∞ E^{q+1}(t) dt ≤ cE(s) using the energy identity and then applies Komornik's lemma as an external decay criterion; the stated polynomial and exponential rates follow from that lemma. Self-citations appear: [26] is cited for the technical time-dependent-δ Young estimate in (20)-(21), and [12], [13], [15] are background or methodological; none carries a central premise whose truth is assumed from the present authors' prior work. The paper's real weakness is not circularity but completeness: Definition 7 defines strong solutions and states 'See [11] for the well-posedness of a similar problems,' whereas [11] treats a linear constant-exponent distributed-delay wave equation, not the variable-exponent problem (8). Local existence, uniqueness, and the asserted regularity for (8) are not proved in the paper, so Theorems 15, 18, and 21 are conditional on a solution class whose existence is assumed. This is a correctness/completeness risk, not a circularity: the conclusions are not equivalent by construction to the hypotheses, and the blow-up time bound is a consequence, not a restatement, of the energy estimates.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on several external results: standard variable-exponent space properties (cited from [19]-[22]), technical lemmas stated without proof from [22], the assumed local existence and regularity of strong solutions (via citation to [11]), and Komornik's lemma. None of these introduce fitted parameters; they are background assumptions.

assumptions (4)

domain assumption Strong solutions of (8) exist with regularity u ∈ C^2([0,T);L^2) ∩ C^1([0,T);H^1_0) ∩ C([0,T);H^2∩H^1_0) and ut ∈ L^{m(·)}(Ω×(0,T))
Definition 7 posits such solutions; the paper cites [11] for well-posedness of a similar problem but does not prove it here. All energy identities and multiplier integrations rely on this regularity.
standard math Lemmas 9-14 from [22] hold (embedding and convexity estimates for variable-exponent spaces)
Stated without proof in Section 4, used to bound H^{α(m2-1)}∫|u|^m dx and to control norms in the blow-up argument.
standard math The log-Hölder condition (3) guarantees density and Sobolev embedding properties used in the proofs
Assumed in (3) and used to justify Poincaré and embedding inequalities in variable-exponent spaces.
standard math Komornik's lemma (Lemma 19) is valid as stated
Taken from [28]; it converts the integral inequality into decay rates.

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Pith. "Pith review of On the wave equation with variable exponent nonlinearity and distributive delay." pith.science (2026). https://pith.science/paper/LIO7AXMY

@misc{pith2026241115528,
  author       = {Pith},
  title        = {Pith review of: On the wave equation with variable exponent nonlinearity and distributive delay},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LIO7AXMY}},
  note         = {Machine review of arXiv:2411.15528}
}

read the original abstract

In this work, we are concerned with a nonlinear wave equation with variable exponents. A distributive delay is imposed into the damping term with variable exponents nonlinearity. Firstly, we show that the global nonexistence time can be dominated. Secondly, global existence of solutions is shown under some suitable conditions on the initial data. Finally, the decay rates of that solutions are established as well.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

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Pith tools

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

[1] [11]

Nicaise and C

S. Nicaise and C. Pignotti, Stabilization of the wave equation with b ound- ary or internal distributed delay, Diﬀ. Int. Equ. 21(9-10) (2008) :935-958

work page 2008

[2] [1]

Levine, Some additional remarks on the nonexistence of glob al solu- tions to nonlinear wave equations, SIAM J

H.A. Levine, Some additional remarks on the nonexistence of glob al solu- tions to nonlinear wave equations, SIAM J. Math. Anal. 5 (1) (1974) :138– 146

work page 1974

[3] [2]

Kopackova, Remarks on bounded solutions of a semilinear dissip ative hyperbolic equation, Comment

M. Kopackova, Remarks on bounded solutions of a semilinear dissip ative hyperbolic equation, Comment. Math. Univ. Carolin. 30 (4) (1989):7 13– 719

work page 1989

[4] [3]

Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch

E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal. 149 (2) (1999):155–182

work page 1999

[5] [4]

Levine, J

H.A. Levine, J. Serrin, Global nonexistence theorems for quasilin ear evo- lution equations with dissipation, Arch. Ration. Mech. Anal. 137 (4) (1997):341–361

work page 1997

[6] [5]

Messaoudi, Blow up in a nonlinearly damped wave equation, Math

S.A. Messaoudi, Blow up in a nonlinearly damped wave equation, Math . Nachr. 231 (1) (2001):1–7

work page 2001

[7] [6]

Zuazua, Exponential decay for the semi-linear wave equation with locally distributed damping, Comm

E. Zuazua, Exponential decay for the semi-linear wave equation with locally distributed damping, Comm. Partial Diﬀerential Equations 15 (1990 ):205- 235

work page 1990

[8] [7]

L. Sun, B. Guo, W. J. Gao, A lower bound for the blow-up time to a damped semilinear wave equation, Appl. Math. Lett. 37(2014):22-2 5

work page 2014