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For the logarithmic heat equation, a weighted Osgood condition forces every nonnegative solution to blow up before the linear terminal time, while its failure gives a sharp amplitude threshold—dividing powers p=2 and p=3/2.

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For the logarithmic Laplacian heat equation, the critical nonlinear growth below which small data survive to the linear terminal time is quadratic for ordinary tails and 3/2-power for critical tails.

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load-bearing objection A novel Fujita-type lifespan theory for the logarithmic Laplacian with tail-dependent critical exponents, but the threshold dichotomy relies on a false bound in Lemma 2.7. the 2 major comments →

arxiv 2607.29318 v1 pith:HVAXIE6Q submitted 2026-07-31 math.AP

Sharp Lifespan Dichotomies and Threshold Phenomena for Semilinear Heat Equations Driven by the Logarithmic Laplacian

Huyuan Chen , Rui Chen , Daniel Hauer , Jun Wang This is my paper
classification math.AP MSC 35K5835A0135B4435R11
keywords logarithmic Laplaciansemilinear heat equationblow-uplifespanOsgood conditionthreshold phenomenonmild solutionsnonlocal diffusion
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 nonnegative mild solutions of ∂ₜu + (−∆)^{ln}u = f(u) with initial data μu₀, where the logarithmic heat kernel exists only for 0 < t < N/2 and is not integrable at spatial infinity. Because the linear flow itself can become singular at a terminal time set by the decay of u₀, the natural question is whether the nonlinearity shortens that lifespan. The main contribution is a complete dichotomy: if a weighted Osgood tail condition holds, every nontrivial solution blows up strictly before the linear terminal time; if it fails (under a mild complementarity condition on f), there is a sharp amplitude threshold μ* separating full linear lifespan for small μ from premature blow-up for large μ. In the critical-tail regime u₀ ≍ (1+|x|)^{−N}, the same dichotomy holds with a square-root weighted condition, so the borderline power drops from p=2 to p=3/2 for f(s)=s^p. This replaces the classical critical-exponent picture with a tail-dependent, scaling-free criterion.

Core claim

The central claim is a complete lifespan dichotomy. Define the weighted Osgood tail Φ(ρ)=ρ∫_ρ^∞ dσ/f(σ). In the slow-decay and fast-decay regimes, if liminf_{ρ→∞} Φ(ρ)=0 then every nonnegative mild solution has maximal existence time T_{μ,f}<T_{μ,0} for every μ>0; if instead liminf Φ(ρ)>0 and f satisfies (F∞), there exists a finite threshold μ* such that T_{μ,f}=T_{μ,0} for μ<μ* and T_{μ,f}<T_{μ,0} for μ>μ*, with T_{μ,f}→0 as μ→∞. For critical-tail initial data u₀≍(1+|x|)^{−N}, the same dichotomy holds with ρ^{1/2} in place of ρ. For the model nonlinearity f(s)=s^p these conditions reduce to p>2 (noncritical) and p>3/2 (critical). The paper also establishes that ∫_{0+} dσ/f(σ)<∞ forbids any

What carries the argument

The central object is the logarithmic heat kernel P_ln(t,x)=P₀(t)|x|^{2t−N}, which is positive and convolutionally usable only for 0<t<N/2, with coefficient P₀(t)∼(N−2t)^{−1} near t=N/2. The linear flow S_ln(t)u₀ has maximal lifespan T_{μ,0}=min{α,N}/2 when u₀ decays like (1+|x|)^{−α}, and its terminal profile is captured by explicit supersolution profiles Φ_γ(t,x)=(γ−2t)^{−1}(1+|x|)^{−(γ−2t)} (γ=N or α) and by the critical profile (1+(N−2t)log(1+|x|))(N−2t)^{−2}(1+|x|)^{−(N−2t)}. The engine is a pair of estimates: weighted convolution bounds and time-integral bounds that allow quadratic (respectively 3/2-power) Duhamel terms to be absorbed into the linear profile, together with a compact-se

Load-bearing premise

The load-bearing premise is Lemma 2.7's bound ∫_a^N b^{−p}P₀((b−a)/2)(1/(b−a)+1/(a+(p−1)b))db ≤ C a^{−1} with C independent of a; its displayed proof asserts P₀(r/2)≤Cr uniformly on (0,N−a), which fails near r=N−a because P₀(t)∼(N−2t)^{−1} as t→N/2, and this lemma is what makes the quadratic supersolutions of Lemma 4.2 work, so the small-amplitude half of Theorem 1.3(b) depends on a correct version.

What would settle it

Compute the integral in (2.17) numerically for N=1, p=2, and a→0 using the exact P₀(t)=π^{−1/2}4^{−t}Γ((1−2t)/2)/Γ(t). If the integral grows faster than C a^{−1} (for instance like a^{−1} log(1/a) or a^{−2}), then the small-amplitude full-lifespan conclusion in Theorem 1.3(b) fails for some f satisfying (F∞); a direct check is whether the proposed supersolution W_{A,N} actually dominates the Duhamel term as a→0.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the dichotomy is correct, any genuinely superlinear source with a sufficiently slow-growing tail will shorten the lifespan for every amplitude, while slower sources admit a sharp amplitude threshold μ* below which the full linear lifespan is attained.
  • For power nonlinearities the thresholds are explicit: p>2 in the noncritical regimes and p>3/2 in the critical-tail regime force premature blow-up for all μ, whereas 1<p≤2 (or 1<p≤3/2) yields a threshold phenomenon.
  • The instantaneous nonexistence theorem implies that sublinear powers such as f(u)=u^p with 0<p<1 admit no nontrivial local solution at all for the logarithmic heat equation, in contrast to the classical heat equation.
  • Matching terminal lower bounds (T−t)^{−1} in noncritical regimes and (T−t)^{−2} in the critical regime with the Osgood upper bound gives sharp blow-up rates, with the borderline exponents p=2 and p=3/2.

Where Pith is reading between the lines

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

  • If the corrected version of the key time-integral bound holds, the same supersolution strategy should extend to nonlinearities slightly above quadratic growth, yielding quantitative estimates for μ* in terms of the Osgood tail; the paper does not compute such rates.
  • The dichotomy suggests that for nonlocal operators with nonintegrable kernels and finite-time singular linear flows, the critical nonlinearity is set by the kernel's blow-up rate at the terminal time, not by a universal scaling exponent—a pattern that could be tested by modifying the kernel's singularity.
  • The p=3/2 critical borderline invites a type I/type II blow-up classification near that exponent, using the linear lower bound (T−t)^{−2}; the paper itself notes this as a natural follow-up.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies the semilinear heat equation with the logarithmic Laplacian, ∂_t u + (-Δ)^ln u = f(u), with initial datum μ u_0. It first shows that the positive logarithmic heat kernel exists only for 0<t<N/2 and that the linear lifespan depends on the spatial decay of u_0. Under the standing assumptions (U_α) and (F), the authors develop a profile-based well-posedness theory, prove instantaneous nonexistence under the Osgood condition at zero, and establish two complementary lifespan criteria. The central results are Theorems 1.3 and 1.4: in the noncritical decay regimes, the weighted Osgood tail condition liminf ρ∫_ρ^∞ dσ/f(σ)=0 forces premature blow-up for every μ>0, while its failure together with (F_∞) yields a sharp amplitude threshold μ* separating full linear lifespan from premature blow-up; in the critical-tail regime the same dichotomy holds with the square-root weighted Osgood condition and a threshold μ*_crit. Theorem 1.5 gives terminal-time lower and upper bounds, including sharp blow-up rates for power nonlinearities.

Significance. If the proofs are completed, this is a substantial contribution. The logarithmic heat semigroup has no standard L^1–L^∞ smoothing, its positive kernel exists only up to N/2, and the linear flow may itself become singular; the paper replaces the classical Fujita critical exponent by tail-dependent Osgood conditions. The profile-based function spaces X_{τ,u_0}, the explicit kernel asymptotics (2.1)–(2.2), the weighted convolution estimates, and the supersolution constructions are natural and largely carefully executed. The paper also makes explicit falsifiable predictions for power nonlinearities (p=2 and p=3/2 dividing powers). However, the proof of Lemma 2.7 contains a false uniform bound that is load-bearing for the noncritical threshold theorem, so the manuscript needs a substantive repair before the central dichotomy can be accepted as proved.

major comments (2)
  1. [Lemma 2.7, §2.2, Eq. (2.17)] The proof of (2.17) uses the uniform estimate P_0(r/2) ≤ C r for all r∈(0,N−a). This is incompatible with the asymptotic (2.2): for t=(b−a)/2, N−2t = N−b+a, so as r=b−a approaches N−a, P_0(r/2) ≍ (N−r)^{-1}, which for small a is of order a^{-1}, not O(r). Thus the displayed bound does not control the upper part of the integration range. This is not cosmetic: Lemma 4.2 invokes Lemma 2.7 with p=2 to obtain the factor Ca^{-1} that yields the quadratic convolution estimates (4.12)–(4.13), and those estimates are the key input for the small-amplitude full-lifespan half of Theorem 1.3(b). The lemma may be repairable using the two-sided bound P_0(t) ≤ C t/(N−2t), but the correction is not present in the manuscript. As written, the noncritical threshold dichotomy is not fully established.
  2. [Theorem 1.4(b), proof, large-data step] In the proof of Theorem 1.4(b), the large-data conclusion is justified by appealing to (1.9), but (1.9) is not explicitly verified in the critical regime before it is used. It does follow from (F_∞), since Φ_f(ρ) ≤ C ρ/f(ρ) < ∞ for large ρ, but the manuscript should state this. This is a derivation omission rather than a fatal gap, but it should be fixed for clarity.
minor comments (4)
  1. [Lemma 2.7, proof, p=1 case] The sentence 'Since the functional n(N/a) is bounded on (0,N)' is inaccurate: ln(N/a) is unbounded as a↓0. The desired bound J_1 ≤ C a^{-1} still follows from ln(N/a) ≤ N/a, but the written justification should be corrected.
  2. [Lemma 2.7, statement] The constant C in (2.17) is said to depend on 'N, p and C_0', but C_0 is not defined in the lemma. If it refers to the constants in the P_0 asymptotics, this should be stated explicitly.
  3. [Lemma 4.2, statement] There is a typo: 'where where' should be 'where'.
  4. [Remark 1.5, piecewise definition of f] The piecewise definition of f has a formatting error: '1,0≤s≤1' should read '1, 0≤s≤1'.

Circularity Check

0 steps flagged

No significant circularity: the lifespan dichotomy is derived, not defined into existence; thresholds and exponents 2 and 3/2 come from independent estimates. Separate flags: background self-citation of [6] (kernel/semigroup facts) and a proof gap in Lemma 2.7 (false P0(r/2)≤Cr bound) that is a correctness risk, not circularity.

full rationale

Central derivation is not circular. The weighted Osgood conditions (1.12) and (1.14) are intrinsic criteria on f alone; for f(s)=s^p they reduce by direct algebra to p>2 and p>3/2, with no lifespan quantity fed back into them. The thresholds mu* in Theorems 1.3(b)/1.4(b) are defined as the supremum of G := {mu>0 : T_{mu,f}=T0}; the dichotomy then rests on three independently proven facts: (i) G is nonempty (small-amplitude supersolutions via Lemmas 4.1-4.2 and 5.1-5.2, with quadratic/3/2-growth bounds derived from failure of the Osgood condition plus (F_infty), cf. (4.21), (5.14)); (ii) G is bounded above (Osgood blow-up, (4.19)); (iii) monotonicity of mu -> T_{mu,f} (Lemma 3.2). The step 'if mu>mu* then T<T0' is definitional only after (i)-(ii) supply the content. The dividing powers 2 and 3/2 are derived, not imposed: they come from matching the linear terminal lower bounds of Lemma 2.5, (N-2t)^{-1} and (N-2t)^{-2}, with the scalar-ODE upper bound Psi_f of Theorem 1.5; the exponent 1/2 in (1.14) is forced by the critical linear growth (N-2t)^{-2} (rho~(N-2t)^{-2} makes Phi_f(rho)<=C(N-2t) equivalent to rho^{1/2}Phi_f(rho)->0). Imported background: kernel formula, positivity window 0<t<N/2, and local semigroup identity are 'taken from [6]', a paper sharing author H. Chen; these are published background facts, the kernel is stated explicitly with asymptotics (2.1)-(2.2) re-derived in the paper, and no lifespan claim reduces to the citation, so per rule 4 this self-citation is not circular. Caveats weighed here: (a) correctness risk, not circularity - the proof of Lemma 2.7 (Eq. 2.17) claims 'P0(r/2)<=Cr' uniformly for r in (0,N-a), which conflicts with the paper's own (2.2) near r=N-a where P0(r/2)~C/(N-r)~C/a; since Lemma 4.2 and the small-amplitude half of Theorem 1.3(b) depend on (2.17), the dichotomy is not fully established as written. (b) The manuscript's Remarks 1.4-1.5 openly describe the profile-norm vs uniform blow-up distinction and the ODE-type matching, consistent with an honest derivation rather than concealment.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No fitted parameters and no new physical entities are introduced. The contribution is a set of existence/blow-up dichotomies under explicit hypotheses on u0 and f. The main background input is the logarithmic kernel theory of [6], a shared-author prior result, plus standard Osgood and rearrangement tools. One internal lemma (Lemma 2.7) has a proof gap that should be repaired.

axioms (5)
  • domain assumption Logarithmic heat kernel formula P_ln(t,x)=P0(t)|x|^{2t-N}, positivity only for 0<t<N/2, and the local semigroup property S_ln(t−s)S_ln(s)=S_ln(t).
    Imported from [6] at the start of Section 2.1; used in (2.6), Theorem 1.2, Lemma 2.2, Lemma 2.3, and all Duhamel restarting arguments. Not re-derived here.
  • domain assumption Standing hypotheses (Uα) and (F): u0 nonnegative, nontrivial, in L∞_α; f locally Lipschitz, nondecreasing, f(0)=0.
    All main theorems are conditional on these assumptions; Theorem 1.1 separately covers non-Lipschitz sublinear f like u^p, p<1.
  • standard math Osgood ODE comparison principle: for v'=κf(v), v(a)=A>0, finiteness of ∫ dσ/f(σ) implies finite-time blow-up.
    Used in Theorem 1.1, Lemma 3.3, Theorem 1.3, and Theorem 1.4 to convert lower bounds into lifespan estimates.
  • standard math Riesz rearrangement inequality for radially nonincreasing functions.
    Used in Lemma 2.4 to obtain the linear lifespan lower bound T_{μ,0}≥Tα.
  • ad hoc to paper Lemma 2.7's bound (2.17) holds with constant independent of a.
    This internal lemma is load-bearing for Lemma 4.2 and Theorem 1.3(b), but its displayed proof uses the false bound P0(r/2)≤Cr near the terminal time; the statement appears repairable.

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Pith. "Pith review of Sharp Lifespan Dichotomies and Threshold Phenomena for Semilinear Heat Equations Driven by the Logarithmic Laplacian." pith.science (2026). https://pith.science/paper/HVAXIE6Q

@misc{pith2026260729318,
  author       = {Pith},
  title        = {Pith review of: Sharp Lifespan Dichotomies and Threshold Phenomena for Semilinear Heat Equations Driven by the Logarithmic Laplacian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HVAXIE6Q}},
  note         = {Machine review of arXiv:2607.29318}
}
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read the original abstract

We study nonnegative mild solutions of \[ \partial_tu+(-\Delta)^{\ln}u=f(u) \qquad \text{in }(0,T)\times\mathbb R^N, \] with initial datum \(u(0,\cdot)=\mu u_0\), \(\mu>0\). Unlike the classical and fractional heat kernels, the positive logarithmic heat kernel exists only for \(0<t<N/2\). The associated linear evolution may become singular at its terminal time, and both its maximal lifespan and terminal growth depend on the spatial decay of \(u_0\). The behavior of \(f\) near zero determines local solvability: if $\int_{0^+}\frac{d\sigma}{f(\sigma)}<\infty,$ then no finite nonnegative solution exists on any positive time interval. Under \((\mathcal U_\alpha)\) and \((\mathcal F)\), we develop a well-posedness theory adapted to the nonintegrable logarithmic heat kernel. We also prove two complementary lifespan criteria: globally at most linear growth of \(f\) yields the full linear lifespan, whereas $\int_{s_*}^{\infty}\frac{d\sigma}{f(\sigma)}<\infty$ implies that the maximal existence time tends to zero as \(\mu\to\infty\). We then establish lifespan dichotomies in the slow-decay, fast-decay, and critical-tail regimes. In the noncritical regimes, the weighted Osgood tail condition forces premature blow-up, while its failure, under \((\mathcal F_\infty)\), yields an amplitude threshold. For critical-tail data, an analogous dichotomy holds with the square-root weighted Osgood condition, and the dividing power becomes \(3/2\). Finally, we derive terminal-time estimates and sharp blow-up rates for power nonlinearities.

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This paper was first reviewed by deepseek-v4-flash on August 3, 2026.