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arxiv: 2606.26555 · v1 · pith:4UCRLKZJnew · submitted 2026-06-25 · 🧮 math.AP

Sharp Lifespan Estimates and Fujita Phenomena for Fractional Hardy-H\'enon Type Parabolic Equations

Mohamed Majdoub , Berikbol T. Torebek This is my paper

Pith reviewed 2026-06-26 04:30 UTC · model grok-4.3

classification 🧮 math.AP
keywords fractional parabolic equationsHardy-Hénon weightFujita exponentlifespan estimatesblow-up phenomenanonlocal operatorssemilinear equationscritical exponents
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The pith

The lifespan of small solutions to the fractional Hardy-Hénon parabolic equation obeys power-law, exponential, or infinite estimates separated by the critical exponent p_F = 1 + (2s - γ)/N.

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

The paper proves sharp estimates for the lifespan of mild solutions to the Cauchy problem u_t + (-Δ)^s u = |x|^{-γ} |u|^p with small positive initial data ε u_0. For the critical exponent p_F = 1 + (2s - γ)/N the lifespan is infinite, while below it the lifespan scales as ε to the power -1/β and at criticality it grows exponentially in ε to the power -(p-1). These results are obtained using fractional heat kernel estimates for the lower bound and testing against the backward fractional heat kernel for the upper bound. A sympathetic reader would care because this determines the precise boundary between global existence and finite-time blow-up for nonlocal parabolic equations with singular weights.

Core claim

We prove that the lifespan T_ε obeys, for every sufficiently small ε>0, T_ε ≈ ε^{-β^{-1}} if 1<p<p_F, exp(C ε^{-(p-1)}) if p=p_F, +∞ if p>p_F, with p_F=1+(2s-γ)/N and β=[(2s-γ)-N(p-1)]/[2s(p-1)]. The lower bound rests on fractional heat-kernel estimates and an L^1-L^∞ Hardy-type interpolation inequality; the upper bound is obtained by testing the equation against the backward fractional heat kernel, a globally defined positive weight for which (-Δ)^s is controlled everywhere and the linear terms cancel identically by self-adjointness. The exponent β is sharp; for γ=0 it reduces to the fractional Lee-Ni exponent.

What carries the argument

Testing against the backward fractional heat kernel, a globally defined positive weight that controls (-Δ)^s everywhere and cancels linear terms by self-adjointness, which produces the upper bound without compactly supported cutoffs.

If this is right

  • The exponent β is sharp and reduces to the fractional Lee-Ni exponent when γ=0.
  • Large-data lifespan laws hold with the same critical exponent.
  • Sharp lower bounds on the blow-up rate hold, together with a conditional Type-I upper bound.
  • A conditional self-similar profile result follows for the blow-up solutions.

Where Pith is reading between the lines

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

  • The backward-kernel testing method may apply to other nonlocal operators whose heat kernels satisfy similar positivity and self-adjoint cancellation properties.
  • Numerical verification of the exact power and exponential rates for concrete s, γ, N would provide independent evidence for the sharpness of β.
  • The same lifespan dichotomy may hold for equations with more general radial weights that preserve the L^1 positivity of the initial data.

Load-bearing premise

The backward fractional heat kernel is a globally defined positive weight for which (-Δ)^s is controlled everywhere and the linear terms cancel identically by self-adjointness.

What would settle it

A numerical computation of the lifespan for a sequence of small ε at p = p_F that fails to exhibit the claimed double-exponential growth rate would falsify the critical case.

read the original abstract

We study the lifespan of mild solutions to the fractional semilinear parabolic Cauchy problem with a Hardy--H\'enon-type weight \[ u_t + (-\Delta)^s u = |x|^{-\gamma}\,|u|^p, \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N, \qquad u(0,x)=\varepsilon\,u_0(x), \] where $0<s<1$, $0\le\gamma<\min(2s,N)$, $p>1$ and $u_0\in L^1\cap L^\infty$ with $\int_{\mathbb{R}^N}u_0(x)\,dx>0$. Setting \[ p_F \;:=\; 1+\frac{2s-\gamma}{N}, \] we prove that the lifespan $T_\varepsilon$ obeys, for every sufficiently small $\varepsilon>0$, \[ T_\varepsilon \;\approx\; \begin{cases} \varepsilon^{-\,\beta^{-1}},& 1<p<p_F,\\[1mm] \exp\!\big(C\,\varepsilon^{-(p-1)}\big),& p=p_F,\\[1mm] +\infty,& p>p_F, \end{cases} \qquad \beta \;=\;\frac{(2s-\gamma)-N(p-1)}{2s(p-1)}. \] The lower bound rests on fractional heat-kernel estimates and an $L^1$--$L^\infty$ Hardy-type interpolation inequality; the upper bound is obtained by testing the equation against the backward fractional heat kernel, a globally defined positive weight for which $(-\Delta)^s$ is controlled everywhere and the linear terms cancel identically by self-adjointness. This circumvents the compactly supported cutoffs of the classical test-function method, which are incompatible with a nonlocal operator. The exponent $\beta$ is sharp; for $\gamma=0$ it reduces to the fractional Lee--Ni exponent $\frac{1}{p-1}-\frac{N}{2s}$. To the best of our knowledge, these results are new even for $\gamma=0.$ We also establish a large-data lifespan law, sharp lower bounds on the blow-up rate together with a conditional Type-I upper bound, a conditional self-similar profile result.

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.

Referee Report

2 major / 2 minor

Summary. The paper studies the lifespan T_ε of mild solutions to the fractional semilinear parabolic problem u_t + (-Δ)^s u = |x|^{-γ} |u|^p on (0,∞)×ℝ^N with small initial data ε u_0 (u_0 ∈ L^1 ∩ L^∞, positive mass). It identifies the Fujita exponent p_F = 1 + (2s - γ)/N and proves the sharp lifespan law T_ε ≈ ε^{-β^{-1}} for 1 < p < p_F, T_ε ≈ exp(C ε^{-(p-1)}) for p = p_F, and T_ε = +∞ for p > p_F, where β = [(2s - γ) - N(p-1)] / [2s(p-1)]. The lower bound uses fractional heat-kernel estimates together with an L^1-L^∞ Hardy-type interpolation; the upper bound tests the equation against a globally defined positive backward fractional heat kernel. Additional results include large-data lifespan, sharp lower bounds on blow-up rate, a conditional Type-I upper bound, and a conditional self-similar profile. The exponent β reduces to the known fractional Lee-Ni value when γ = 0.

Significance. If the derivations hold, the work supplies a complete small-data Fujita classification for the nonlocal parabolic equation with Hardy-Hénon weight, extending the γ = 0 case. The backward-kernel testing method is a technical contribution that circumvents compact-cutoff difficulties for nonlocal operators. The paper also ships sharp blow-up-rate lower bounds and conditional profile results. These are new even for γ = 0 and rest on standard but carefully adapted tools (heat kernels, self-adjointness).

major comments (2)
  1. [upper-bound paragraph (abstract) and §4] The abstract states that the backward fractional heat kernel is globally defined, positive, and satisfies the required control on (-Δ)^s with exact cancellation of linear terms by self-adjointness, but the manuscript must verify that this kernel remains a valid test function for the weighted nonlinearity |x|^{-γ} |u|^p when γ > 0 (the weight may affect positivity or integrability at infinity).
  2. [Introduction, paragraph after the lifespan law] The claimed reduction of β to the fractional Lee-Ni exponent 1/(p-1) - N/(2s) when γ = 0 is algebraically correct, but the manuscript should explicitly record the resulting lifespan exponent 1/β and confirm it matches the literature statement used for comparison.
minor comments (2)
  1. [Abstract, lifespan display] Notation: the symbol ≈ in the lifespan statement should be replaced by explicit two-sided inequalities with constants independent of ε (or at least stated as T_ε ∼ c ε^{-β^{-1}} with c > 0).
  2. [Introduction] The large-data lifespan law and the conditional Type-I / self-similar results are announced but their precise statements (including any extra assumptions on u_0) should appear in the introduction for quick reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive suggestions. We address each major comment below and will make the indicated revisions to improve clarity.

read point-by-point responses

  1. Referee: [upper-bound paragraph (abstract) and §4] The abstract states that the backward fractional heat kernel is globally defined, positive, and satisfies the required control on (-Δ)^s with exact cancellation of linear terms by self-adjointness, but the manuscript must verify that this kernel remains a valid test function for the weighted nonlinearity |x|^{-γ} |u|^p when γ > 0 (the weight may affect positivity or integrability at infinity).

    Authors: We agree that an explicit verification strengthens the presentation. In the revised manuscript we will add a short paragraph in §4 (immediately after the definition of the backward kernel) confirming that, under the standing assumption 0 ≤ γ < min(2s,N), the kernel φ remains positive, the integral ∫ |x|^{-γ} |u|^p φ dx is well-defined for the mild solutions under consideration, and the self-adjoint cancellation of the linear terms continues to hold. The argument relies only on the known positivity and decay properties of the fractional heat kernel together with the local integrability of |x|^{-γ} near the origin and its decay at infinity; no change to the proofs is required. revision: yes

  2. Referee: [Introduction, paragraph after the lifespan law] The claimed reduction of β to the fractional Lee-Ni exponent 1/(p-1) - N/(2s) when γ = 0 is algebraically correct, but the manuscript should explicitly record the resulting lifespan exponent 1/β and confirm it matches the literature statement used for comparison.

    Authors: We accept the suggestion. In the revised introduction we will insert, immediately after the sentence on the reduction of β, the explicit formula 1/β = 2s(p-1)/(2s - N(p-1)) and state that the resulting lifespan law T_ε ≈ ε^{-1/β} coincides with the known small-data lifespan for the fractional semilinear heat equation (γ=0) as obtained by Lee-Ni and subsequent works. This is a purely expository addition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The lifespan estimates are obtained from explicit constructions: lower bounds via fractional heat-kernel estimates plus L1-L∞ Hardy interpolation, upper bounds via testing against the globally defined backward fractional heat kernel (using self-adjointness to cancel linear terms). The exponent β is defined directly from the parameters (2s-γ, N, p) with no fitting step or renaming of an input quantity. The reduction to the Lee-Ni case when γ=0 is a consistency check, not a load-bearing premise. No self-citations, ansatzes smuggled via citation, or self-definitional loops appear in the derivation chain. The argument is independent of the target lifespan law.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claim rests on standard properties of the fractional heat kernel, self-adjointness of (-Δ)^s, and existence of mild solutions; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (2)
  • standard math Fractional heat kernel estimates and L1-L∞ Hardy-type interpolation inequality hold for the given range of s and γ.
    Invoked explicitly for the lower bound in the abstract.
  • standard math Backward fractional heat kernel is globally positive and (-Δ)^s is controlled with linear terms canceling by self-adjointness.
    Invoked explicitly for the upper bound in the abstract.

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Reference graph

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