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REVIEW 4 major objections 4 minor 21 references

Blow-up solutions of parabolic $p$-Laplacian inequalities on locally finite graphs

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that on infinite locally finite graphs, any nonzero nonnegative solution of the parabolic p-Laplacian inequality with a source growing at least like $u^{p-1}$ blows up in finite time.

desk verdict Main theorem is false as stated because the lower bound on σ is not uniform in time, but the comparison principle is sound and the flaw is fixable. read the letter →

arxiv 2507.12303 v2 pith:KXXNLU7T submitted 2025-07-16 math.AP

Wenyuan Ma , Liang Zhao This is my paper · ORCID
classification math.AP MSC 35R0235B4435K55
keywords parabolicinequalityfinite-timeblow-uplocallyfinitegraphp-LaplaciancomparisonprincipleDirichletboundaryfirsteigenvalueBernoullisubsolution
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 establishes that nonnegative, nontrivial solutions of the parabolic $p$-Laplacian inequality $u_t - \Delta_p u \geq \sigma(x,t)\Phi(u)$ on an infinite locally finite graph must blow up in finite time under two sets of conditions: if $\Phi(s) \geq (C_0+\varepsilon_0)s^{p-1}$ with $C_0$ tied to a lower bound of $\sigma$, or if $\Phi(s) \geq C_1 s^q$ with $1

What carries the argument

The paper's central tool is the comparison principle (Lemma 2.3) for the parabolic $p$-Laplacian inequality on a finite connected subgraph with zero Dirichlet boundary. It is proved by multiplying the two competing solutions by $e^{-\lambda t}$, taking a minimum point of the difference, and using the monotonicity of the function $g(a)=|a|^{p-2}a$. This principle lets the authors transplant blow-up from an auxiliary finite-graph problem (2.4) to the original infinite graph. The superlinear blow-up in Lemma 3.1 rests on the maximum function $m(t)=\max_{x\in U} u(x,t)$, whose time derivative is controlled by the inequality $m_t \geq \delta\Phi(m)-m^{p-1}$. The subcritical case in Lemma 3.2 uses the first eigenvalue $\lambda_1$ and positive eigenfunction $\varphi_1$ of the discrete $p$-Laplacian on $U$, defined via the Rayleigh quotient and a Lagrange multiplier argument (Lemma 2.4), to build an explicit subsolution of Bernoulli type.

What would settle it

Solve, numerically or analytically, the finite-graph Dirichlet problem (2.4) on a path graph with $\sigma\equiv 1$, $p=3$, $\Phi(u)=u^2$ (power $p-1$ with no $\varepsilon_0$ slack), and constant initial data $u_0\equiv 1$; if a global solution exists, the strict inequality in Theorem 1.1 is essential, whereas finite-time blow-up would show the threshold is not sharp.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: if $\Phi(s) \geq (C_0+\varepsilon_0)s^{p-1}$ for $s\geq 0$, where $C_0$ is the reciprocal of a positive lower bound of $\sigma$ on a finite connected subgraph and $\varepsilon_0>0$, then for every nonnegative initial value $u_0$ not identically zero, any solution of (1.3) blows up in finite time at every vertex with $u_0(x_0)>0$. The proof fixes a finite connected subgraph $U$ containing such a vertex, and uses the comparison principle to show that the restriction of $u$ to $U$ dominates the solution $v$ of an auxiliary Dirichlet problem. For $v$, the spatial maximum $m(t)=\max_{x\in U} v(x,t)$ is nondecreasing and satisfies $m_t \geq \delta\Phi(m) - m^{p-1} > 0$, which integrates to $t \leq F(m(0))$ with $F$ finite, forcing blow-up before time $F(m(0))$. Theorem 1.2 covers slower power growth $\Phi(s) \geq C_1 s^q$, $1<q\leq p-1$, by constructing a subsolution $v=h(t)\varphi_1(x)$ from a Bernoulli ODE whose solution blows up, provided the initial data is at least $h_0$.

Load-bearing premise

The proof needs the potential $\sigma(x,t)$ to stay above a fixed positive constant $\delta$ on the chosen finite subgraph for the whole existence time; if $\sigma$ decays to zero in time, the constant $C_0$ in Theorem 1.1 cannot be fixed and the key inequality $\delta\Phi(s)-s^{p-1}>0$ can fail.

Editorial extensions

If this is right

  • Every nonzero nonnegative solution with $\Phi(s) \geq (C_0+\varepsilon_0)s^{p-1}$ explodes at each vertex where the initial value is positive, regardless of how small that value is.
  • The blow-up time is bounded by $F(u_0)=\int_{u_0}^{\infty} \frac{ds}{\delta\Phi(s)-s^{p-1}}$, a finite number that depends only on the vertex maximum of the initial data and the lower bound of $\sigma$.
  • For power growth $C_1 s^q$ with $1<q\leq p-1$, sufficiently large initial data, quantified explicitly by $h_0$, forces blow-up even though the growth is below the $p-1$ threshold.
  • The comparison principle gives a general recipe: prove a qualitative property on a finite subgraph with Dirichlet boundary, then lift it to any locally finite graph.

Reading between the lines

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

  • The borderline case $\Phi(s)=C_0 s^{p-1}$ with no $\varepsilon_0$ slack is not covered; whether arbitrary small nonzero data still blow up there would test whether the strict inequality is an artifact of the proof.
  • The Bernoulli-subsolution construction should extend to coupled systems of $p$-Laplacian inequalities, where the single ODE for $h$ becomes a system and blow-up can be diagnosed by eigenvalue comparisons.
  • A numerical phase diagram in the $(q, \|u_0\|_{\ell^\infty})$ plane on an infinite path graph could reveal whether the exponent $p-1$ is the true critical Fujita exponent for locally finite graphs, and whether the large-data threshold $h_0$ is close to sharp.
  • The comparison principle as stated needs zero Dirichlet boundary; a version with non-zero boundary data would let one localize blow-up to regions where the solution is already large, which is useful for inhomogeneous graphs.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. This paper studies the parabolic p-Laplacian inequality (1.3) on locally finite connected weighted graphs. It proves local existence via a Banach fixed-point argument, establishes a comparison principle on finite vertex subsets, and then uses subsolutions built from ODEs and the first Dirichlet eigenfunction to claim finite-time blow-up: Theorem 1.1 for Φ growing at least like s^{p-1}, and Theorem 1.2 for Φ ≳ s^q with 1 < q ≤ p-1 and sufficiently large initial data. The main tools are comparison with solutions on finite subsets and an eigenfunction-based Bernoulli subsolution.

Significance. Blow-up criteria for semilinear heat equations on graphs are an active topic, and a treatment of the parabolic p-Laplacian inequality with a time-dependent potential would be a reasonable contribution. The local existence argument and the comparison principle are useful building blocks. However, the central blow-up theorems are not supported: Theorem 1.1 is false as stated (there is a global solution satisfying all stated hypotheses), and the proof of Theorem 1.2 contains a homogeneity error. These issues are load-bearing, so the paper cannot be accepted in its present form.

major comments (4)
  1. [§3, Lemma 3.1 / Theorem 1.1] The proof introduces δ as the minimum of σ on U × [0, T′] for an arbitrary T′ < T, where T is the unknown maximal existence time, and then takes C0 = δ^{-1}. This does not produce a fixed constant: different choices of T′ give different δ, and no uniform-in-time lower bound on σ is assumed. Consequently, the inequalities (3.1)–(3.3) are integrated with a δ that is not fixed on the whole interval of existence. The theorem is false without such a bound. On G = Z with p = 3, take σ(x,t) = e^{-t}, Φ(s) = 10s^2, and u0 ≡ y0 with 0 < y0 < 0.1. Conditions (i)–(iv) hold and (1.4) holds with C0 = 9 and ε0 = 0.5, but the spatially constant solution y(t) = (y0^{-1} - 10(1 - e^{-t}))^{-1} solves (1.3) as an equality and is global. The theorem needs an explicit hypothesis such as inf_{U×[0,∞)} σ ≥ δ > 0, and the proof must use that fixed δ.
  2. [§3, Lemma 3.2 / Eqs. (3.5)–(3.6)] The construction of the subsolution v = h(t)φ_1(x) is invalid because the p-Laplacian is homogeneous of degree p-1, not degree one. The correct identity is Δ_{p|U}(h(t)φ_1) = h(t)^{p-1}Δ_{p|U}φ_1 = -λ_1 h(t)^{p-1}|φ_1|^{p-2}φ_1, whereas (3.6) uses the linear expression -λ_1 h(t)|φ_1|^{p-2}φ_1. For h ≥ 1 the true diffusion term is more negative than the one used, so the inequality (3.6) does not follow and the Bernoulli ODE (3.5) does not provide a valid subsolution. This invalidates the proof of Theorem 1.2; additionally, the constant δ used in (3.5) is never defined in Lemma 3.2.
  3. [§2, Corollary 2.2 / Theorem 1.2] The proof of Theorem 1.2 chooses U = {x0}. For such a set, the Rayleigh quotient in Lemma 2.4 is identically zero because there are no edges inside U, so the argument that λ1 > 0 for connected induced graphs does not apply to the singleton case used in the theorem. The Dirichlet eigenvalue problem on a singleton does have a positive eigenvalue if the boundary edges are included (λ = deg(x0)/μ(x0)), but that value is not obtained by the Rayleigh quotient given in the paper. This is another gap in the proof of Theorem 1.2.
  4. [§3, proof of Theorem 1.1] The proof restricts the full-graph solution u to U and assumes u(x,t) ≥ 0 on ∂U × [0,T) in order to apply Lemma 2.3 to the auxiliary problem (2.4). Nonnegativity of solutions of the inequality (1.3) for nonnegative initial data is not established anywhere; Lemma 2.1 only gives local existence and continuity in time. Without a maximum principle for the inequality on the whole graph, the application of (1.4) for s ≥ 0 and the boundary comparison are not justified.
minor comments (4)
  1. [§3, Lemma 3.2] The statement of Lemma 3.2 uses the symbol v both for the solution of (2.4) and for the subsolution h(t)φ_1(x), which makes the statement and proof difficult to follow.
  2. [§2, Corollary 2.2] There is a typo 'on on' in the first sentence, and in the definition of φε the expression 'φ(x)' should read 'φ1(x)'.
  3. [§2, Lemma 2.3] The comparison principle assumes u, v ∈ C^1([0,T)) in time, but the existence lemmas only provide continuity in time; the regularity needed for the blow-up arguments should be stated and proved.
  4. [§2, Lemma 2.1] In the proof of Lemma 2.1, the sentence 'D[v](x,t) ∈ C[0,T] for x ∈ U' should refer to V rather than U, since the lemma concerns the whole vertex set.

Circularity Check

2 steps flagged · score 6.0 of 10

The main theorems are not independent of the conclusion: the constants C0 and h0 appearing in their hypotheses are defined in the proofs using the unknown maximal existence time, making the assumptions self-referential.

  1. self definitional [Section 3, Lemma 3.1 and Remark 1.1 (proof of Theorem 1.1)]
    "By assumptions (i) and (ii) in Section 1, there exists δ > 0 such that σ(x, t) ≥ δ on U × [0, T′]. By (1.4), we can take C0 = δ−1 such that δΦ(s) − s^{p−1} > (δC0 − 1)s^{p−1} = 0 for s ≥ 0. ... The constant C0 in Theorem 1.1 can be given a specific value in our proof of the theorem."

    Here T′ is an arbitrary endpoint of the interval [0,T′) on which the solution is assumed to exist. Choosing δ as the minimum of σ over U × [0,T′] makes C0 = δ^{-1} a function of the unknown existence time T, which is precisely the quantity Theorem 1.1 must show is finite. The hypothesis (1.4) is therefore not a fixed condition on σ, u0 and Φ alone; it is manufactured from the lifespan being analyzed. For a globally existing solution with inf_{t≥0} σ = 0, such as σ = e^{-t}, the required C0 grows without bound as T′ → ∞, so the assumed inequality can hold only for solutions whose maximal existence time is already bounded.

  2. self definitional [Section 3, Eq. (3.5) and Lemma 3.2 (proof of Theorem 1.2)]
    "Set φmin := min_{x∈U} φ > 0. ... h(0) = h0 := (λ1∥φ1∥ℓ∞(U)/(C1δφ^q_min))^{1/(q−1)} + ε. ... Assume min_{x∈U} u0 is sufficiently large and let v be a solution of (2.4)."

    The constant h0 is defined using δ, which in the proof must be a positive lower bound of σ on the existence interval of the solution, exactly as in Lemma 3.1. Hence the condition '∥u0∥ℓ∞(V) sufficiently large', namely ∥u0∥ℓ∞(V) ≥ h0, depends on the unknown maximal existence time of the very solution whose blow-up is being proved. No uniform-in-time lower bound for σ is assumed, so for a decaying σ the threshold h0 may be infinite on global solutions, making the hypothesis unverifiable or vacuous. The result is thus conditional on a bound on the target quantity rather than on a fixed data-only assumption.

full rationale

The paper contains no fitted parameters, no data predictions, and no load-bearing self-citations: the cited results by Chung–Choi, Mastrolia–Monticelli–Punzo, and others are used as standard tools or inspiration, and the one self-reference (Yang–Zhao) is only an introductory citation. However, the two main theorems have a self-referential structure in their hypotheses. In Theorem 1.1, the constant C0 is announced as 'determined by σ and u0', but the proof of Lemma 3.1 defines C0 = δ^{-1} with δ = min σ over U × [0,T′], where T′ is tied to the solution's lifespan. This means the assumption (1.4) is not an a priori condition on the data; it encodes the conclusion that the lifespan is finite. The same issue appears in Theorem 1.2, where the threshold h0 for 'sufficiently large' initial data is defined through the same time-interval-dependent δ. Consequently, the proof does not establish blow-up from fixed hypotheses; it establishes a conditional statement whose hypotheses are only knowable after knowing the maximal existence time. This is a genuine self-definitional circularity in the central arguments, though it is not a case of renaming known results or smuggling an ansatz by citation. Because the main theorems' hypotheses reduce, by construction, to a bound on the quantity being proved finite, a score of 6 is appropriate.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new particles or structures. Its main theoretical burden is the unstated uniform lower bound on σ and the ill-defined constants C0 and h0. The rest relies on standard analytic tools: fixed point theorems, Lagrange multipliers, and Bernoulli ODE blow-up.

free parameters (2)
  • C0 = δ^(-1) = δ^(-1), where δ = min_{U x [0,T']} σ(x,t) for a finite interval
    In Lemma 3.1 and Theorem 1.1, C0 is set to δ^(-1) after choosing a finite interval [0,T']. Because σ's lower bound may decay with the interval length, C0 is not a fixed a priori constant; the theorem's hypothesis is ill-posed without a uniform lower bound on σ.
  • h0 in Theorem 1.2 = (λ1 ||φ1||^(p-2)/(C1 δ φ_min^q))^(1/(q-1)) + ε
    The threshold h0 depends on δ, a lower bound of σ over an unspecified time interval. The 'sufficiently large' condition is therefore not explicit in terms of known data unless a global lower bound on σ is assumed.
assumptions (7)
  • standard math Banach fixed-point theorem
    Used in Lemmas 2.1 and 2.2 to prove local existence of solutions.
  • standard math Lagrange multiplier theorem
    Used in Lemma 2.4 to derive the eigenvalue equation from the constrained minimization of the Rayleigh quotient.
  • standard math Bernoulli equation blow-up criterion
    Used in Theorem 1.2 to show that h(t) in (3.5) blows up in finite time for the chosen initial condition.
  • domain assumption p > 2 and locally Lipschitz Φ
    The function g(a) = |a|^(p-2)a is locally Lipschitz and strictly increasing only under p > 2; the comparison principle relies on this.
  • domain assumption σ positive and continuous in t, bounded in x
    Assumptions (i) and (ii) in the introduction; needed for local existence and for the existence of δ on finite intervals.
  • ad hoc to paper Uniform positive lower bound of σ on U x [0,∞)
    Not stated in the paper but required for Lemma 3.1 and Theorem 1.1 to hold. The proof chooses δ over a finite interval depending on the solution's lifespan; without a uniform lower bound, the inequality δΦ(s) - s^(p-1) > 0 can fail at large times.
  • ad hoc to paper Nonnegativity of solutions for nonnegative initial data
    The comparison argument requires u ≥ 0 on the boundary and in the domain. The paper states u0 ≥ 0 but does not prove that an arbitrary solution of the inequality remains nonnegative.

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Cite this review

Pith. "Pith review of Blow-up solutions of parabolic $p$-Laplacian inequalities on locally finite graphs." pith.science (2026). https://pith.science/paper/KXXNLU7T

@misc{pith2026250712303,
  author       = {Pith},
  title        = {Pith review of: Blow-up solutions of parabolic $p$-Laplacian inequalities on locally finite graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KXXNLU7T}},
  note         = {Machine review of arXiv:2507.12303}
}
abstract

In this paper, we study blow up behavior of the semilinear parabolic inequality with $p$-Laplacian operator and nonlinear source $u_t - \Delta_p u \geq \sigma(x, t)\Phi(u)$ on a locally finite connected weighted graph $G = (V, E)$. We extend the comparison principle and thereby establish the relationship between the initial value and the existence of blow-up solutions to the problem under different growth rates of $\Phi$. We prove that when the growth rate of $\Phi$ exceeds linear growth, blow-up solutions exist under appropriate initial conditions.

Discussion (0). Continue with ORCID to comment.

Reference graph

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