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arxiv: 2605.24802 · v1 · pith:B63Z55MOnew · submitted 2026-05-24 · 🧮 math.AP

Global existence and blow-up for the Hardy-Sobolev parabolic equation in RN

Fei Fang , Zhong Tan This is my paper

Pith reviewed 2026-06-30 00:20 UTC · model grok-4.3

classification 🧮 math.AP
keywords Hardy-Sobolev parabolic equationself-similar transformationglobal existencefinite-time blow-uppotential well methodPalais-Smale conditionweighted Hardy inequality
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The pith

A self-similar transformation reduces the Hardy-Sobolev parabolic equation to one whose solutions are classified as globally existing or finite-time blowing up via the modified potential well method.

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

The authors apply a self-similar change of variables to the original parabolic equation containing a Hardy term, producing an equivalent equation that includes a linear drift term. They first establish a weighted Hardy inequality that makes the associated energy functional well-defined. They then apply the modified potential well method together with Palais-Smale sequence analysis to obtain criteria that separate initial data leading to global solutions from those that blow up in finite time.

Core claim

After the self-similar transformation, the modified potential well method partitions the phase space into invariant sets separated by the mountain-pass level of the energy functional, so that solutions either exist for all positive times or blow up in finite time according to whether the initial datum lies inside or outside the potential well.

What carries the argument

The modified potential well method applied after the self-similar transformation, which uses the mountain-pass geometry and Palais-Smale compactness of the energy functional in the presence of the weighted Hardy term.

Load-bearing premise

The transformed equation admits an energy functional whose mountain-pass geometry and Palais-Smale compactness hold for the stated range of the Hardy parameter μ.

What would settle it

An explicit initial datum whose energy lies below the mountain-pass level yet produces a solution that blows up in finite time, or whose energy lies above the level yet produces a global solution, would falsify the classification.

read the original abstract

In this paper, we apply a self-similar transformation to convert the parabolic equation with a Hardy term \begin{equation*} \begin{cases}u_t-\Delta u-\mu \frac{u}{|x|^2}=|u|^{2^*-2} u & \text { in } \mathbb{R}^N \times(0, T), u(x, 0)=u_0(x) & \text { in } \mathbb{R}^N , \end{cases} \end{equation*} into the following parabolic equation \begin{equation*} \begin{cases} v_s-\Delta v-\frac{1}{2} y \cdot \nabla v=\beta v+\frac{\mu v}{|y|^2}+|v|^{2^*-2} v &\text { in } \mathbb{R}^N \times(0, S), \left.v\right|_{s=0}=v_0 & \text { in } \mathbb{R}^N, \end{cases} \end{equation*} where $N \geqslant 3$, $\mu\in [0,(N-2)^2 /8]$ and $2^{\ast}=2N /(N-2)$. For this equation, we establish a weighted Hardy inequality. Furthermore, by virtue of the modified potential well method and Palais-Smale sequence analysis, we investigate the long-time behavior and finite-time blow-up properties of solutions to the parabolic equation.

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 / 1 minor

Summary. The paper applies a self-similar change of variables to transform the Hardy-Sobolev parabolic equation u_t - Δu - μ u/|x|^2 = |u|^{2*-2}u into an equivalent equation with drift and linear terms, proves a weighted Hardy inequality for μ ∈ [0,(N-2)^2/8], and then uses the modified potential-well method together with Palais-Smale sequence analysis to classify global existence versus finite-time blow-up of solutions.

Significance. If the technical steps hold, the work supplies a systematic extension of the potential-well technique to parabolic problems with singular Hardy potentials, yielding sharp criteria for global existence and blow-up that are not available from the standard Sobolev case alone.

major comments (1)
  1. The abstract invokes the weighted Hardy inequality and subsequent verification of mountain-pass geometry plus Palais-Smale compactness for the transformed energy, yet no explicit statement appears of the precise form of the inequality or the compactness argument; without these details the central claim that the potential-well method applies directly cannot be verified.
minor comments (1)
  1. The range μ ∈ [0,(N-2)^2/8] is correctly identified as the interval where the quadratic form remains positive, but the manuscript should state the constant in the weighted Hardy inequality explicitly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment. We respond to the major comment below.

read point-by-point responses

  1. Referee: The abstract invokes the weighted Hardy inequality and subsequent verification of mountain-pass geometry plus Palais-Smale compactness for the transformed energy, yet no explicit statement appears of the precise form of the inequality or the compactness argument; without these details the central claim that the potential-well method applies directly cannot be verified.

    Authors: We agree that the abstract, as a concise overview, does not include the explicit mathematical form of the weighted Hardy inequality or the details of the Palais-Smale compactness verification. These elements are developed in full in the body of the paper. To address the referee's concern and improve clarity, we will revise the abstract to state the precise form of the weighted Hardy inequality (for μ in the given range) and note that the mountain-pass geometry and Palais-Smale condition are verified for the transformed energy functional. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins with a standard self-similar change of variables that produces the transformed equation with its linear drift and scaling terms as direct artifacts of the substitution; this is not a self-definition but a routine rescaling. The paper then states that it establishes a weighted Hardy inequality for the given range of μ and applies the modified potential-well method together with Palais-Smale compactness. Both the inequality and the functional-analytic tools are presented as independently verifiable steps whose validity does not reduce to any fitted parameter or prior result defined in terms of the target conclusion. No load-bearing self-citation, uniqueness theorem imported from the authors, or ansatz smuggled via citation appears in the provided chain. The central claims therefore remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard Sobolev embeddings, the classical Hardy inequality in the admissible range of μ, and the validity of the modified potential-well geometry after the self-similar change of variables. No free parameters or invented entities are visible from the abstract.

axioms (2)
  • standard math Sobolev embedding H^1(R^N) → L^{2^*}(R^N) holds for N ≥ 3
    Required for the critical nonlinearity and the energy functional
  • domain assumption The weighted Hardy inequality established in the paper controls the singular term for μ in [0, (N-2)^2/8]
    Invoked immediately after the self-similar transformation

pith-pipeline@v0.9.1-grok · 5788 in / 1439 out tokens · 36319 ms · 2026-06-30T00:20:56.289617+00:00 · methodology

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