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arxiv: 2407.13085 · v2 · submitted 2024-07-18 · 🧮 math.AP

On the Hardy-H\'enon heat equation with an inverse square potential

Divyang G. Bhimani , Saikatul Haque , Masahiro Ikeda This is my paper

Pith reviewed 2026-05-23 23:09 UTC · model grok-4.3

classification 🧮 math.AP
keywords Hardy-Hénon equationinverse square potentialheat semigroupweighted Lebesgue spaceslocal well-posednessfinite time blow-upFujita exponentsemilinear parabolic equation
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The pith

Sharp fixed-time decay estimates for the heat semigroup with inverse square potential enable well-posedness and blow-up results for the Hardy-Hénon equation.

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

The paper proves sharp decay estimates for the semigroup generated by minus the Laplacian plus a times one over x squared, acting in weighted Lebesgue spaces. These linear estimates are applied to the semilinear Hardy-Hénon equation with a power nonlinearity multiplied by a power of the distance to the origin. The estimates yield local well-posedness in subcritical and critical weighted spaces, small-data global existence in the critical case, finite-time blow-up for some data in the subcritical regime, and nonexistence of local positive weak solutions when the power exceeds the Fujita exponent in the supercritical regime.

Core claim

We establish sharp fixed time decay estimates for heat semigroups e to the minus t times open parenthesis negative Laplacian plus a over x squared close parenthesis in weighted Lebesgue spaces. As an application, we establish local well-posedness in scale subcritical and critical weighted Lebesgue spaces and small data global existence in critical weighted Lebesgue spaces. Further, under certain conditions on gamma and alpha, we show that local solution cannot be extended to global one for certain initial data in the subcritical regime. We also demonstrate nonexistence of local positive weak solution in supercritical case for alpha greater than one plus open parenthesis two plus gamma close,

What carries the argument

The heat semigroup generated by the operator negative Laplacian plus a over x squared, which supplies the sharp decay rates in weighted Lebesgue spaces that close the estimates for the nonlinear problem.

If this is right

  • Local well-posedness holds in scale-subcritical and critical weighted Lebesgue spaces.
  • Small-data global existence holds in the critical weighted Lebesgue spaces.
  • Local solutions in the subcritical regime fail to extend globally for certain initial data, producing finite-time blow-up.
  • No local positive weak solutions exist in the supercritical regime when alpha exceeds one plus open parenthesis two plus gamma close parenthesis over d.

Where Pith is reading between the lines

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

  • The decay estimates could be tested numerically by evolving initial data under the linear equation and measuring the weighted norms over time.
  • The same linear machinery might apply to other power weights or to systems with multiple components.
  • The Fujita exponent identified here separates existence from nonexistence and could be compared directly with known exponents for the standard heat equation without the inverse-square term.

Load-bearing premise

The linear heat semigroup generated by negative Laplacian plus a over x squared satisfies the claimed sharp decay estimates in the weighted Lebesgue spaces for a greater than or equal to minus open parenthesis d minus two over two close parenthesis squared.

What would settle it

A calculation or explicit example showing that the semigroup decay rate in some weighted Lebesgue space differs from the stated sharp rate for an admissible value of a would falsify the linear estimates and thereby the nonlinear conclusions.

Figures

Figures reproduced from arXiv: 2407.13085 by Divyang G. Bhimani, Masahiro Ikeda, Saikatul Haque.

Figure 1
Figure 1. Figure 1: Local well-posedness in L q s(Rd) occurs in the deep & medium dark region by Theorem 1.2 (only the boundary τ = τc is included). Uniqueness in mere L q s(Rd) is guaranteed by Theorem 1.2 (furthermore part) in the open deep dark region. No LWP in the unbounded lightest regionby Theorem 1.4. – Theorem 1.2 recover results mentioned in Remark 1.3(1a) and is the main part of a detailed well-posedness Theorem 4.… view at source ↗
read the original abstract

We study Cauchy problem for the Hardy-H\'enon parabolic equation with an inverse square potential, namely, \[\partial_tu -\Delta u+a|x|^{-2} u= |x|^{\gamma} F_{\alpha}(u),\] where $a\ge-(\frac{d-2}{2})^2,$ $\gamma\in \mathbb R$, $\alpha>1$ and $F_{\alpha}(u)=\mu |u|^{\alpha-1}u, \mu|u|^\alpha$ or $\mu u^\alpha$, $\mu\in \{-1,0,1\}$. We establish sharp fixed time-time decay estimates for heat semigroups $e^{-t (-\Delta + a|x|^{-2})}$ in weighted Lebesgue spaces. This may be of independent interest. As an application, we establish local well-posedness in scale subcritical and critical weighted Lebesgue spaces and small data global existence in critical weighted Lebesgue spaces. Further, under certain conditions on $\gamma$ and $\alpha,$ we show that local solution cannot be extended to global one for certain initial data in the subcritical regime. Thus, finite time blow-up in the subcritical Lebesgue space norm is exhibited. We also demonstrate nonexistence of local positive weak solution (and hence failure of local well-posedness) in supercritical case for $\alpha>1+\frac{2+\gamma}{d}$ the Fujita exponent.

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

1 major / 2 minor

Summary. The paper establishes sharp fixed-time decay estimates for the heat semigroup e^{-t(-Δ + a|x|^{-2})} in weighted Lebesgue spaces for a ≥ -((d-2)/2)^2. These are applied to obtain local well-posedness in scale-subcritical and critical weighted Lebesgue spaces, small-data global existence in the critical case, finite-time blow-up in the subcritical regime for certain data under conditions on γ and α, and nonexistence of local positive weak solutions in the supercritical regime when α > 1 + (2+γ)/d (the Fujita exponent). Three forms of the nonlinearity F_α are treated.

Significance. If the linear decay estimates hold sharply across the full range including the Hardy threshold, the results would extend known decay and well-posedness theory for the heat equation and Hardy potential to the Hénon-type setting with weights, providing a reasonably complete local/global/blow-up picture in critical spaces. The linear estimates themselves would be of independent interest.

major comments (1)
  1. [Linear estimates section (Theorem on semigroup bounds, likely §2)] Linear estimates section (Theorem on semigroup bounds, likely §2): The decay estimates are asserted for the full range a ≥ -((d-2)/2)^2, but at equality the quadratic form is only non-negative (vanishing on |x|^{-(d-2)/2}). It is not clear from the argument whether a separate justification is given for the boundary case or whether the proof implicitly assumes a positive lower bound on the spectrum or reduces to the free Laplacian via a shift that becomes zero only at equality. This is load-bearing for the sharpness claim and therefore for all nonlinear conclusions.
minor comments (2)
  1. [Abstract] Abstract: 'fixed time-time decay estimates' contains a repeated word; should be 'fixed-time decay estimates'.
  2. [Introduction / main results] Notation for the three cases of F_α is introduced but the subsequent statements on well-posedness and blow-up do not always distinguish which case is being treated; a short clarifying sentence or table would help.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the need for explicit treatment of the boundary case in the linear estimates. We address the single major comment below.

read point-by-point responses

  1. Referee: Linear estimates section (Theorem on semigroup bounds, likely §2): The decay estimates are asserted for the full range a ≥ -((d-2)/2)^2, but at equality the quadratic form is only non-negative (vanishing on |x|^{-(d-2)/2}). It is not clear from the argument whether a separate justification is given for the boundary case or whether the proof implicitly assumes a positive lower bound on the spectrum or reduces to the free Laplacian via a shift that becomes zero only at equality. This is load-bearing for the sharpness claim and therefore for all nonlinear conclusions.

    Authors: We agree that the boundary case a = -((d-2)/2)^2 requires explicit justification to support the sharpness claim. The proof of Theorem 2.1 proceeds via the quadratic form associated to the operator and the resulting spectral gap for a > -((d-2)/2)^2. At equality the form is nonnegative but the gap vanishes. The estimates nevertheless hold by a limiting argument: the constants appearing in the decay bounds are continuous with respect to a down to the threshold, and the same weighted estimates follow by density from the strict inequality case. Equivalently, the critical Hardy operator is unitarily equivalent (via the weight |x|^{-(d-2)/2}) to a nonnegative perturbation of the free Laplacian whose heat kernel satisfies the same weighted bounds. We will revise Section 2 by adding a dedicated remark (or short subsection) that spells out this limiting procedure and the unitary equivalence, thereby removing any ambiguity. The nonlinear conclusions remain unchanged. revision: yes

Circularity Check

0 steps flagged

No circularity: linear estimates derived independently before nonlinear application

full rationale

The paper first asserts and (presumably) proves sharp fixed-time decay estimates for the linear semigroup e^{-t(-Δ + a|x|^{-2})} in weighted Lebesgue spaces for a ≥ -((d-2)/2)^2, treating these as an independent result of possible separate interest. These estimates are then applied to obtain local well-posedness, small-data global existence, finite-time blow-up, and nonexistence statements for the nonlinear Hardy-Hénon equation. No equation, parameter fit, or self-citation reduces any claimed prediction or conclusion back to the nonlinear results by construction; the derivation chain begins with the linear operator analysis and proceeds outward without self-referential loops or renaming of known patterns as new derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the linear semigroup estimates whose proof details are absent from the abstract; the only explicit structural assumption visible is the lower bound on a that makes the quadratic form nonnegative.

axioms (1)
  • standard math The operator -Δ + a|x|^{-2} generates a positive semigroup when a ≥ -((d-2)/2)^2
    This is the critical Hardy constant ensuring non-negativity of the associated quadratic form; it is invoked to define the linear evolution.

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