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$L^p$-Asymptotic Profiles for the Heat Equation with a Hardy Potential

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Pith's one-line read Higher-order L^p profiles for the Hardy heat equation are built from the small-argument expansion of the modified Bessel function in the radial kernel, with remainder decay improving by one power of t per order.

desk verdict Clean higher-order L^p profiles for the Hardy heat equation, built directly from the Bessel expansion; solid and self-contained. read the letter →

arxiv 2607.07171 v2 pith:PWZQEHEX submitted 2026-07-08 math.AP

Radu Ordean This is my paper · ORCID
classification math.AP MSC 35K0535B4035C2033C10
keywords HardypotentialheatequationasymptoticprofilesmodifiedBesselfunctionsphericalharmonicsL^pdecayweightedmoments
verification ladder T0 review T1 audit T2 compute T3 formal

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 the large-time behaviour of the heat equation with an inverse-square Hardy potential. For radial initial data with enough weighted moments, it builds explicit asymptotic profiles A_n by expanding the modified Bessel function that appears in the radial Hardy heat kernel to order n. After subtracting A_n, the remainder decays in L^p like t to the power minus (gamma_p + n), where gamma_p is an explicit exponent that reduces to the classical Gaussian rate when the potential vanishes. Each extra term therefore buys one extra power of decay. Non-radial data are handled by expanding in spherical harmonics: every angular mode satisfies its own radial Hardy equation with a shifted parameter, so the same profiles can be summed, finitely or (under a summability condition) infinitely. The construction recovers the classical heat asymptotics when the potential is zero and earlier L^2 results in the critical Hardy case.

What carries the argument

The explicit profile A_n obtained by collecting all terms of total order j <= n in the joint small-argument expansion of the modified Bessel factor I_nu(z) and the Gaussian factor e^{-|y|^2/4t} inside the radial Hardy heat kernel; each collected term is a weighted moment times a Gaussian-type function of |x|^2/t.

What would settle it

Take a radial datum whose first n weighted moments vanish but whose (n+1)-st moment is nonzero and finite; check whether ||S_lambda(t)u_0||_{L^p} is asymptotically exactly C t^{-gamma_p-n} rather than o(t^{-gamma_p-n}).

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Extended reading notes

Core claim

For radial data with finite weighted L^1_{nu,n} norm, the mild solution generated by the radial Hardy heat kernel satisfies lim t^{gamma_p + n} ||u(.,t) - A_n(.,t)||_{L^p} = 0 for p in a range that depends on n and the Hardy parameters; the profile A_n is assembled from the first n+1 weighted moments via the small-z expansion of I_nu. The same expansion, applied mode by mode, yields finite and infinite angular versions for non-radial data.

Load-bearing premise

The initial datum must have enough weighted moments of order n (or n+1 for the remainder estimates); without them the expansion method cannot control the remainder at the claimed rate.

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Editorial analysis

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

Referee Report

0 major / 5 minor

Summary. The paper constructs explicit higher-order asymptotic profiles A_n for mild solutions of the heat equation with Hardy potential on R^N (N≥3, 0≤λ≤λ*). For radial data in the weighted space L^1_{ u,n}, A_n is assembled from the first n+1 weighted moments M_{ u,j}(u_0) via the small-argument expansion of the modified Bessel function I_ u appearing in the radial Hardy heat kernel; the remainder satisfies t^{γ_p+n}‖u(·,t)-A_n(·,t)‖_{L^p} o0 as t o∞ for p in a range determined by local integrability near the origin and large-z kernel estimates. The same expansion is applied mode-by-mode after spherical-harmonic decomposition to obtain finite and (under summability) infinite angular expansions of the profile. Sharpness of the rate, recovery of the classical Gaussian case (λ=0) and of the L^2 result of Vázquez–Zuazua, and a self-contained derivation of the radial/non-radial kernels via Hankel transform are included.

Significance. If correct, the work supplies the first explicit higher-order L^p asymptotic profiles for the Hardy heat equation, improving on the leading-term results of Vázquez–Zuazua, Pilarczyk and Cazacu–Ignat–Manea by an arbitrary number of powers of t^{-1}. The profiles are completely explicit (polynomials in |x|^2/t times the weighted moments), the remainder estimates are elementary once the kernel expansion is written, and the argument recovers the classical heat-equation asymptotics as a special case. The spherical-harmonic reduction and the density argument via a dual moment basis are clean and reusable. These features make the paper a useful reference for large-time analysis of singular parabolic equations.

minor comments (5)
  1. In the proof of Theorem 2.2 a stray opening quotation mark appears before t^{γ_p+n}; remove it.
  2. Lemma 3.1 records two equivalent expressions for γ_p; keep only the definition (3) and derive the second form once, to avoid visual duplication.
  3. The lower threshold on p in Proposition 3.2 is written with +4 while Theorem 2.1 uses +0; a one-sentence remark that the stronger moment hypothesis of the proposition forces the stricter lower bound would clarify the relation.
  4. In the large-z estimate (Step 4) the constant M is chosen equal to u-μ+2n+2+(N-1)/2; a brief parenthetical note that any larger M also works would make the argument more flexible for the reader.
  5. References [6] and [12] are listed but never cited; either incorporate them or delete them.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: profiles are explicit Taylor remainders of a re-derived kernel; decay rates follow from direct estimates under stated moment hypotheses.

full rationale

The paper constructs A_n by collecting terms of total order j = m + ℓ from the small-argument series of I_ u and the Taylor expansion of the y-Gaussian factor inside the known radial Hardy heat kernel (6); the weighted moments M_{ u,j}(u_0) are then simply the linear functionals that multiply those terms. Proposition 3.2 bounds the four remainder pieces (Bessel, Gaussian, cut-off, large-z) by O(t^{-γ_p-n-1}) once the (n+1)-st moment is finite, and the density argument of Theorem 2.1 upgrades the estimate to the claimed o(t^{-γ_p-n}) under the weaker L^1_{ u,n} hypothesis. The kernel itself is re-derived from first principles in the Appendix via Hankel transform and spherical harmonics, so no external uniqueness theorem or fitted parameter is load-bearing. There is no self-definitional loop, no data-fitting step renamed as prediction, and no uniqueness claim imported from prior work by the same author. The derivation is therefore self-contained against its own stated assumptions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The paper rests entirely on classical special-function identities (modified Bessel expansion, Hankel inversion, spherical-harmonic decomposition of the Laplacian) and on the Hardy inequality that defines the critical constant lambda_*. No free parameters are fitted; the profiles are constructed, not postulated. The only domain assumptions are the range 0 <= lambda <= lambda_* and the weighted integrability of the initial datum.

assumptions (4)
  • standard math Small-argument expansion of the modified Bessel function I_nu(z) = sum_{m=0}^n (z/2)^{2m+nu}/(m! Gamma(m+nu+1)) + R_{I,n}(z) with |R| <= C z^{2n+nu+2} for z <= 2
    Invoked in Step 1 of Proposition 3.2; taken from Watson's treatise.
  • domain assumption Hardy inequality with optimal constant lambda_* = ((N-2)/2)^2
    Defines the admissible range of lambda and the parameter nu; cited from Brezis-Marcus and used throughout the introduction and kernel derivation.
  • standard math Spherical-harmonic decomposition of L^2(S^{N-1}) and the associated eigenvalues ell(ell+N-2)
    Used to reduce the non-radial problem to a family of radial Hardy equations (Proposition 4.2).
  • standard math Hankel inversion formula of order nu and the Weber discontinuous integral that produces the modified-Bessel heat kernel
    Core of the radial kernel derivation in Proposition 4.1; classical identities from Watson and Gradshteyn-Ryzhik.
invented entities (1)
  • Higher-order asymptotic profiles A_n built from weighted moments M_{nu,j} and the polynomials P_j independent evidence
    purpose: To capture the large-time expansion of the solution up to arbitrary order n
    Explicitly constructed from the kernel expansion; not an independent physical object. independent_evidence is true because the profiles are completely determined by the initial datum and the known kernel.

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Pith. "Pith review of $L^p$-Asymptotic Profiles for the Heat Equation with a Hardy Potential." pith.science (2026). https://pith.science/paper/PWZQEHEX

@misc{pith2026260707171,
  author       = {Pith},
  title        = {Pith review of: $L^p$-Asymptotic Profiles for the Heat Equation with a Hardy Potential},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PWZQEHEX}},
  note         = {Machine review of arXiv:2607.07171}
}
abstract

For radial initial data, we construct explicit higher-order \(L^p(\mathbb R^N)\)-asymptotic profiles for the heat equation with Hardy potential. These profiles, denoted $A_n$ are obtained from the small-argument expansion, up to an arbitrary order \(n\), of the modified Bessel function appearing in the radial Hardy heat kernel. If $u$ is the mild solution generated by this kernel, we prove that the corresponding remainder $u(x,t)-A_n(x,t)$ admits a polynomial decay depending on $n$ in \(L^p(\mathbb R^N)\) as \(t\to\infty\). We also treat the non-radial case through spherical harmonics: each angular mode evolves according to a radial Hardy heat equation with a modified parameter, leading to finite and infinite angular expansion versions of the asymptotic profile under suitable summability assumptions.

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Works this paper leans on

21 extracted references · 21 canonical work pages

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