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arxiv: 2604.27516 · v1 · submitted 2026-04-30 · 🧮 math.AP

Barriers, Barenblatt solutions and regularity of soda can domains for the heat equation and nonlinear p-parabolic equations

Anders Bj\"orn , Jana Bj\"orn This is my paper

Pith reviewed 2026-05-07 09:28 UTC · model grok-4.3

classification 🧮 math.AP
keywords soda can domainsboundary regularityp-parabolic equationsheat equationBarenblatt solutionsbarriersnonconvex domainsorigin regularity
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The pith

For the heat equation and p-parabolic equations with p less than 2n/(n+1), the regularity of the origin in soda can domains is completely determined.

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

This paper establishes a complete characterization of when the origin is a regular boundary point for soda can domains with nonconvex power-law time sections under the p-parabolic equation. For all p below 2n over n plus 1 and for the heat equation at p equals 2, explicit criteria distinguish regular from irregular cases at the origin. A sympathetic reader would care because the result extends classical boundary regularity theory from convex time sections to these nonconvex geometries, which better capture certain diffusion processes. The determination relies on constructing barriers and Barenblatt solutions tailored to the power-law dependence in the time slices.

Core claim

We completely determine when the origin is regular for soda can domains of the type Theta sub l comma theta for the p-parabolic equation when p is less than 2n over n plus 1 and for the heat equation p equals 2. The domains have nonconvex time sections with power dependence on the spatial variable, and the regularity status is decided using barriers and Barenblatt solutions.

What carries the argument

Barriers and Barenblatt solutions constructed to test regularity of the origin for the nonconvex soda can domains Theta sub l comma theta.

If this is right

  • The origin is regular precisely when the exponent l lies above or below a critical threshold that depends on p and dimension n.
  • The complete classification holds uniformly for all theta greater than zero.
  • Regularity decisions require no further assumptions beyond the stated range of p and the power-law form of the time sections.
  • The nonconvex case complements earlier complete results for convex time sections.

Where Pith is reading between the lines

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

  • The same barrier technique may apply directly to other families of domains whose time sections deviate from convexity by a power law.
  • Explicit regularity thresholds could be used to construct test cases for numerical schemes solving p-parabolic equations near singular boundary points.
  • Physical models of diffusion in regions whose cross-sections expand or contract according to a power of time now have precise conditions for well-posedness at the last instant.

Load-bearing premise

The barrier and Barenblatt constructions suffice to decide regularity for all nonconvex power-law time sections without additional restrictions on the solution class or domain parameters.

What would settle it

A specific pair of l and p in the considered range together with an explicit solution of the p-parabolic equation that is regular at the origin when the barrier criterion predicts irregularity, or vice versa.

read the original abstract

In this paper we study when the origin $(0,0)$ is a regular (or irregular) boundary point for the so-called soda can domains of the type \[ \Theta_{l,\theta}:= \{(x,t) \in \mathbf{R}^{n+1}: 0<-t < \theta |x|^l <\theta\}, \quad \text{with $l,\theta >0$,} \] for the $p$-parabolic equation $\partial_t u- \Delta_p u=0$, where $1<p<\infty$. For $p<2n/(n+1)$ and for the heat equation (i.e.\ $p=2$) we completely determine when the origin is regular for soda can domains. The domains $\Theta_{l,\theta}$ have nonconvex time sections with power dependence on time. For domains with rotationally symmetric convex time sections with power dependence on time, the regularity of the origin as the last point was characterized by Petrovskii (in 1935) for the heat equation, and almost completely in the nonlinear case ($p \ne 2$) in our earlier paper (joint with Gianazza, Math. Ann. 368 (2017), 885--904).

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

0 major / 3 minor

Summary. The paper studies regularity of the origin (0,0) as a boundary point for nonconvex soda-can domains Θ_{l,θ} = {(x,t) ∈ R^{n+1} : 0 < -t < θ |x|^l < θ} (l,θ > 0) for the p-parabolic equation ∂_t u - Δ_p u = 0 with 1 < p < ∞. For p < 2n/(n+1) and for the heat equation (p=2), it claims a complete determination of when the origin is regular or irregular, via explicit barrier constructions based on Barenblatt profiles (nonlinear case) and tailored supersolutions (linear case) that are verified to satisfy the comparison principle inside these domains; this extends the authors' prior convex-case characterization (with Gianazza, Math. Ann. 2017) and Petrovskii's 1935 result.

Significance. If the direct verifications of the supersolutions hold without hidden restrictions on l, θ or the solution class, the result supplies a full, explicit characterization for this family of nonconvex power-law domains. The approach strengthens the literature by showing that Barenblatt profiles and curvature-controlled computations suffice to handle the nonconvexity uniformly near the origin, providing a concrete model for extending regularity theory beyond convex time sections.

minor comments (3)
  1. [Abstract] Abstract, domain definition: the chained inequality 0 < -t < θ |x|^l < θ is notationally ambiguous (does it impose |x|^l < 1 simultaneously, or is it two separate conditions?). Clarify the precise set membership in the introduction or §2.
  2. [Main construction section] The statement that the constructions are 'valid supersolutions inside the nonconvex soda-can domains' and 'verified directly against the p-parabolic operator' should include a brief remark on the precise range of l for which the time-section curvature remains controlled; even if uniform, an explicit bound would aid readability.
  3. [Comparison principle application] Ensure the comparison principle is stated with the exact hypotheses used (e.g., boundedness or integrability of solutions) when applied to the nonconvex geometry, to avoid any appearance of implicit restrictions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and for recommending minor revision. No specific major comments were listed in the report, so we have no points requiring response or changes.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper determines regularity of the origin for nonconvex soda-can domains Θ_{l,θ} via explicit barrier constructions using Barenblatt profiles (for 1 < p < 2n/(n+1)) and tailored supersolutions (for p=2). These are verified directly by substitution into the p-parabolic operator and application of the comparison principle on the specific power-law geometry, without reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The reference to the authors' 2017 convex-case paper supplies only background context for the extension; the new nonconvex determination rests on independent direct computation of curvature effects near the origin and does not collapse to prior inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard existence and comparison properties of Barenblatt solutions for the p-parabolic equation together with the definition of regularity via barriers. No free parameters are fitted and no new entities are postulated.

axioms (2)
  • standard math Barenblatt solutions exist and serve as comparison functions for the p-parabolic equation
    Invoked to construct barriers; standard in the literature on nonlinear parabolic equations.
  • domain assumption The p-parabolic operator satisfies the usual comparison principle and maximum principle
    Required for barrier method to decide regularity.

pith-pipeline@v0.9.0 · 5532 in / 1184 out tokens · 43578 ms · 2026-05-07T09:28:22.157403+00:00 · methodology

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