arxiv: 2605.09074 · v1 · submitted 2026-05-09 · 🧮 math.AP

Recognition: no theorem link

Double Criticality for a Hardy-Rellich Biharmonic Heat Equation in an Exterior Domain

Bessem Samet, Hadeel Alhatlani, Mohamed Jleli

Pith reviewed 2026-05-12 02:30 UTC · model grok-4.3

classification 🧮 math.AP

keywords biharmonic heat equationHardy-Rellich potentialFujita critical exponentexterior domainexistence and nonexistenceweighted nonlinearitysemilinear parabolic

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The pith

The biharmonic heat equation with singular Hardy-Rellich potential and weighted nonlinearity in an exterior domain has two distinct critical exponents separating existence from nonexistence of weak solutions.

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

This paper determines a Fujita-type critical exponent that marks the boundary between nonexistence and possible existence of weak solutions for the equation. Above that threshold, a second critical exponent appears, governed by the decay rate of the positive source term at infinity. A sympathetic reader cares because these thresholds control whether solutions remain global or cease to exist in models of higher-order diffusion with singular potentials. The analysis incorporates the Hardy-Rellich term and the weight |x|^σ in the nonlinearity, which shifts the critical values compared with the non-singular case. The results therefore refine earlier conclusions on blow-up versus existence for this class of parabolic equations.

Core claim

We identify two distinct critical regimes governing the behavior of solutions. We first determine a Fujita-type critical exponent that separates nonexistence from existence. We then show that, in the supercritical range, a second critical exponent arises in terms of the decay exponent of the source, in the sense of Lee and Ni. Our results extend the recent work by considering a singular Hardy-Rellich potential and a weighted nonlinearity, leading to a different critical behavior.

What carries the argument

The singular Hardy-Rellich potential together with the weighted power nonlinearity |x|^σ |u|^p inside the inhomogeneous biharmonic heat equation on an exterior domain, which together determine the scaling that fixes the two critical exponents.

If this is right

Weak solutions do not exist below the Fujita-type critical exponent regardless of the source term.
In the supercritical range for p, existence or nonexistence is further decided by whether the source decays faster or slower than a second explicit rate.
The presence of the Hardy-Rellich potential changes both critical exponents relative to the corresponding equation without the singular term.
The critical behavior holds for admissible ranges of the weight exponent σ and for sufficiently large space dimension N.

Where Pith is reading between the lines

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

The same pattern of two successive critical exponents may appear in other higher-order parabolic equations that include singular potentials.
The thresholds supply concrete tests that numerical schemes for such equations can be checked against.
Varying the weight parameter σ while keeping other data fixed would produce a continuous family of critical exponents whose dependence could be mapped explicitly.

Load-bearing premise

The domain is an exterior region in high-dimensional space, the potential is singular at the origin, and the source term has a prescribed decay rate at infinity.

What would settle it

The existence of a global weak solution for a power p strictly below the claimed Fujita critical exponent, or for a source whose decay rate lies on the wrong side of the second threshold, would falsify the nonexistence statements.

read the original abstract

We study the existence and nonexistence of weak solutions to an inhomogeneous semilinear biharmonic heat equation in an exterior domain, involving a singular Hardy--Rellich potential, a weighted nonlinearity $|x|^{\sigma}|u|^{p}$, and a positive source term $f(x)$. We identify two distinct critical regimes governing the behavior of solutions. More precisely, we first determine a Fujita-type critical exponent that separates nonexistence from existence. We then show that, in the supercritical range, a second critical exponent arises in terms of the decay exponent of the source, in the sense of Lee and Ni. Our results extend the recent work \cite{Tobakhanov} by considering a singular Hardy--Rellich potential and a weighted nonlinearity, leading to a different critical behavior.

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.

Desk Editor's Note private letter to a colleague

This paper gets a double-criticality result for the biharmonic heat equation by adding the Hardy-Rellich potential and the weighted nonlinearity, which shifts the thresholds from the cited prior work.

read the letter

The main takeaway is that the authors identify two distinct critical regimes for this inhomogeneous semilinear biharmonic heat equation in an exterior domain. A Fujita-type exponent separates global existence from nonexistence, and then in the supercritical range a second threshold appears that depends on the decay rate of the positive source term, following the Lee-Ni pattern. The addition of the singular Hardy-Rellich potential and the |x|^σ |u|^p term produces different critical values than in the Tobakhanov reference, which is the actual novelty here.

Referee Report

2 major / 3 minor

Summary. The manuscript studies the existence and nonexistence of weak solutions to an inhomogeneous semilinear biharmonic heat equation posed in an exterior domain of R^N. The PDE incorporates a singular Hardy-Rellich potential, the weighted nonlinearity |x|^σ |u|^p, and a positive source f(x) with prescribed decay at infinity. The central claims are the identification of a Fujita-type critical exponent that separates global nonexistence from existence, followed by a second critical exponent (in the sense of Lee-Ni) that governs the admissible decay rate of f in the supercritical regime. The results are presented as an extension of prior work that produces qualitatively different critical behavior due to the singular potential and weight.

Significance. If the proofs are correct, the paper supplies a concrete instance of double criticality for a higher-order parabolic equation with singular coefficients. This refines the classical Fujita and Lee-Ni theory by showing how the Hardy-Rellich term and the weight σ alter the thresholds, which is a modest but genuine advance for the analysis of semilinear evolution equations in exterior domains.

major comments (2)

[§2] §2 (Definition of weak solutions): The precise function space and test-function class used to define weak solutions must be stated explicitly, including the integrability requirements near the origin that make the Hardy-Rellich term well-defined. This definition is load-bearing for both the nonexistence argument (via integral identities) and the comparison principle invoked in the supercritical regime.
[Theorem 1.1] Theorem 1.1 (Fujita critical exponent): The derivation of the critical value p_F must be checked against the explicit form of the Hardy-Rellich potential; it is not immediately clear whether the potential merely shifts the exponent or introduces additional restrictions on admissible test functions that could affect the threshold.

minor comments (3)

[Abstract] The abstract should list the admissible ranges for N, σ and p so that the parameter regime is immediately visible.
[Introduction] The introduction would benefit from a short paragraph contrasting the new critical exponents with those obtained in the cited work without the singular potential.
[References] All references, especially the extended work, should appear with complete bibliographic data.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and will incorporate clarifications in the revised manuscript to enhance precision.

read point-by-point responses

Referee: [§2] §2 (Definition of weak solutions): The precise function space and test-function class used to define weak solutions must be stated explicitly, including the integrability requirements near the origin that make the Hardy-Rellich term well-defined. This definition is load-bearing for both the nonexistence argument (via integral identities) and the comparison principle invoked in the supercritical regime.

Authors: We agree that the definition requires greater explicitness. In the revised version, Section 2 will be expanded to state the precise function space (a weighted Sobolev space incorporating the biharmonic operator and Hardy-Rellich potential) and the class of test functions, with explicit integrability conditions near the origin ensuring the potential term is well-defined. This will directly support the integral identities in the nonexistence proof and the comparison principle. revision: yes
Referee: [Theorem 1.1] Theorem 1.1 (Fujita critical exponent): The derivation of the critical value p_F must be checked against the explicit form of the Hardy-Rellich potential; it is not immediately clear whether the potential merely shifts the exponent or introduces additional restrictions on admissible test functions that could affect the threshold.

Authors: We have rechecked the derivation. The Hardy-Rellich potential shifts the value of p_F via the associated linear eigenvalue problem but does not introduce further restrictions on test functions; the admissible class remains the same weighted test functions used in the unperturbed case. We will add a short explanatory remark in the proof of Theorem 1.1 to make this shift explicit and confirm the test-function choice. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from PDE structure

full rationale

The paper's central claims identify Fujita-type and Lee-Ni-type critical exponents for existence/nonexistence and source decay in the supercritical regime. These thresholds are obtained by direct analysis of the biharmonic heat equation with Hardy-Rellich potential and weighted nonlinearity in an exterior domain, using comparison principles and extensions of known results from the cited prior work [Tobakhanov]. No step reduces a claimed prediction to a fitted parameter, self-definition, or load-bearing self-citation chain; the extension to the singular potential and weight is explicitly stated to produce different behavior without circular reduction. The derivation remains independent of the target results themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard tools of parabolic PDE theory (Hardy-Rellich inequality, Sobolev embeddings in exterior domains, comparison principles for weak solutions) that are assumed without re-proof. No new free parameters or invented entities are introduced; the critical exponents are derived rather than fitted.

axioms (2)

standard math The Hardy-Rellich inequality holds for the biharmonic operator in exterior domains
Invoked to control the singular potential term in the weak formulation.
domain assumption Comparison principles and test-function methods apply to weak solutions of the inhomogeneous biharmonic heat equation
Used to separate existence from nonexistence across the critical exponents.

pith-pipeline@v0.9.0 · 5438 in / 1450 out tokens · 60529 ms · 2026-05-12T02:30:55.037503+00:00 · methodology

discussion (0)

Reference graph

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