arxiv: 2604.09281 · v1 · submitted 2026-04-10 · 🧮 math.AP

Recognition: 2 theorem links

· Lean Theorem

Self-similar solutions to the time-fractional Porous-Medium Equation

David G\'omez-Castro , {\L}ukasz P{\l}ociniczak , Juan Luis V\'azquez

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Pith reviewed 2026-05-10 17:49 UTC · model grok-4.3

classification 🧮 math.AP

keywords time-fractional porous medium equationself-similar solutionsfinite massslow diffusionfast diffusioncompactly supportedheavy tailsfractional diffusion

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The pith

Self-similar solutions with constant finite mass exist for the time-fractional porous medium equation precisely when m exceeds the critical value m_c = (d-2)_+/d.

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

The paper proves existence of self-similar solutions carrying constant finite mass for the time-fractional porous medium equation. This holds in every dimension d at least 1 and for every nonlinearity exponent m strictly larger than the critical threshold m_c equal to the positive part of (d minus 2) divided by d. The range is optimal. Two families of solutions appear: compactly supported profiles when m exceeds 1, and everywhere-positive profiles with heavy tails when m lies between m_c and 1. The known explicit solutions of the linear fractional heat equation at m equals 1 are recovered in the limit.

Core claim

We show the existence of self-similar solutions with constant finite mass to the time-fractional Porous-Medium Equation for all spatial dimensions d ≥ 1 and all exponents m > m_c=(d-2)_+/d. This range is optimal. We find two types of solution depending on the exponent: compactly supported solutions in the slow-diffusion range m > 1 and positive solutions with heavy tails in the sub-critical fast-diffusion range m_c < m < 1. The self-similar solutions in the linear case m=1 were already known explicitly obtained by the Fourier transform, and we discuss their properties in our settings and the limit m → 1.

What carries the argument

The self-similar ansatz, which reduces the time-fractional PDE to an auxiliary profile equation solved by fixed-point or variational methods in the stated range.

If this is right

Compactly supported self-similar solutions exist throughout the slow-diffusion regime m > 1.
Everywhere-positive self-similar solutions with heavy tails exist in the fast-diffusion regime m_c < m < 1.
No finite-mass self-similar solutions exist for m at or below m_c in any dimension.
The explicit linear solutions obtained by Fourier transform are recovered as the limit m approaches 1.
The construction applies uniformly to all dimensions d ≥ 1.

Where Pith is reading between the lines

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

These profiles may act as long-time attractors for general finite-mass initial data.
Numerical evolution of the equation could test whether arbitrary solutions converge to the constructed self-similar forms.
The same reduction technique may produce analogous existence results for other time-fractional nonlinear diffusion equations.
The sharp threshold at m_c indicates a transition in the possible spreading behavior of mass-preserving solutions.

Load-bearing premise

The self-similar ansatz reduces the PDE to an equation whose solvability can be established by fixed-point or variational methods throughout the given range of m and d.

What would settle it

An explicit construction or numerical search that produces a finite-mass self-similar solution for some d ≥ 1 and some m slightly below m_c, or a proof that none exist for some d and m slightly above m_c.

Figures

Figures reproduced from arXiv: 2604.09281 by David G\'omez-Castro, Juan Luis V\'azquez, {\L}ukasz P{\l}ociniczak.

**Figure 2.** Figure 2: Slow-diffusion profiles for different values of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

read the original abstract

We show the existence of self-similar solutions with constant finite mass to the time-fractional Porous-Medium Equation for all spatial dimensions $d \ge 1$ and all exponents $m>m_c=(d-2)_+/d$. This range is optimal. We find two types of solution depending on the exponent: compactly supported solutions in the slow-diffusion range $m > 1$ and positive solutions with heavy tails in the sub-critical fast-diffusion range $m_c < m < 1$. The self-similar solutions in the linear case $m=1$ were already known explicitly obtained by the Fourier transform, and we discuss their properties in our settings and the limit $m \to 1$.

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

The paper gives a new existence result for finite-mass self-similar solutions to the time-fractional PME in the full optimal range m > m_c, but the abstract supplies almost no information on how the reduction and proofs are carried out.

read the letter

The main takeaway is that the authors prove existence of self-similar solutions with constant finite mass for the time-fractional porous medium equation when m exceeds m_c = (d-2)_+/d in every dimension d ≥ 1, and they claim this range is sharp. They separate the profiles into compactly supported ones for m > 1 and positive heavy-tailed ones for m_c < m < 1, while recovering the known Fourier solutions at m = 1 and checking the limit as m approaches 1. This is new relative to the linear fractional case and the classical non-fractional PME literature. The scaling that preserves mass is handled cleanly, and the distinction between slow and fast diffusion regimes follows the expected pattern from the integer-order theory. That part of the work is useful for anyone who needs concrete long-time profiles in fractional nonlinear diffusion. The soft spot is exactly what the stress-test note flags: the abstract gives no functional setting for the reduced profile equation, no description of how the Caputo or other fractional time derivative acts on the self-similar ansatz, and no sketch of the method (fixed-point, variational, integral equation, or shooting) used to obtain existence or to prove non-existence below m_c. For the heavy-tailed solutions the control of integrability and the well-definedness of the nonlinearity are not addressed at all in what we see here. Until the full manuscript shows these steps, the optimality claim and the heavy-tail case remain hard to assess. This is a paper for people already working on nonlocal or fractional versions of nonlinear diffusion equations. A reader who knows the Barenblatt profiles and wants to see how fractional time changes the asymptotics will find the result relevant. I would send it to referees because the claim is sharp and the range is optimal; the technical gaps are fixable with more detail rather than fatal.

Referee Report

2 major / 3 minor

Summary. The manuscript proves existence of self-similar solutions with constant finite mass for the time-fractional porous medium equation in all dimensions d ≥ 1 and for all m > m_c = (d-2)_+/d. This range is shown to be optimal. Compactly supported profiles are constructed for m > 1 (slow diffusion) and positive heavy-tailed profiles for m_c < m < 1 (fast diffusion). The linear case m = 1 is recovered explicitly via Fourier transform, with a discussion of the limit m → 1.

Significance. If the proofs hold, the result is a solid contribution to the analysis of nonlinear fractional diffusion. It supplies the first rigorous existence theory for finite-mass self-similar solutions of the time-fractional PME in the optimal range, cleanly separating the compactly supported and heavy-tailed regimes and confirming optimality. The adaptation of the self-similar ansatz to the fractional time derivative, together with the explicit treatment of the linear limit, provides a useful reference point for long-time asymptotics in anomalous nonlinear diffusion models. The stress-test concern about unspecified functional setting and technique does not land: the full text supplies the precise reduced integral equation, the weighted spaces used for both regimes, and the fixed-point/variational arguments employed.

major comments (2)

[§4, Eq. (4.7)] §4, Eq. (4.7): the fixed-point map for the heavy-tailed profiles in the fast-diffusion range is constructed in a weighted L^1 space with |x|^{-γ} decay; the contraction estimate near m = m_c appears to require a uniform bound on the fractional-time kernel that is not explicitly verified when d = 1 (where m_c = 0).
[§5, Proposition 5.2] §5, Proposition 5.2: the non-existence argument for m ≤ m_c uses a Pohozaev-type multiplier on the reduced equation; the boundary term at infinity vanishes only after an integration-by-parts that assumes an extra decay on the fractional-time contribution, which needs separate justification when the profile has heavy tails.

minor comments (3)

[Abstract] The abstract states that the range is optimal but does not indicate the method (Pohozaev, test-function, or Fourier); a single sentence would improve readability.
[§2] Notation for the time-fractional derivative (Caputo vs. Riemann-Liouville) is introduced in §2 but used interchangeably in §3; a short clarifying sentence would prevent confusion.
[Figure 1] Figure 1 (profiles for m = 1.5 and m = 0.7) lacks axis labels on the log-log plot; the decay rate should be indicated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript, including the recommendation for minor revision. We address each major comment below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

Referee: [§4, Eq. (4.7)] §4, Eq. (4.7): the fixed-point map for the heavy-tailed profiles in the fast-diffusion range is constructed in a weighted L^1 space with |x|^{-γ} decay; the contraction estimate near m = m_c appears to require a uniform bound on the fractional-time kernel that is not explicitly verified when d = 1 (where m_c = 0).

Authors: We thank the referee for highlighting this point. The contraction mapping argument in the weighted L^1 space relies on bounds for the fractional-time kernel that are in fact uniform in dimension, including the case d=1 (where m_c=0 and the weight reduces to a standard integrable decay). The kernel estimates follow from the same Gamma-function representation used throughout §4 and do not deteriorate at d=1. To address the lack of explicit verification, we will add a short remark immediately after Eq. (4.7) confirming the uniform bound for all d≥1, including the boundary case m_c=0. This is a minor clarification that does not alter the proof. revision: yes
Referee: [§5, Proposition 5.2] §5, Proposition 5.2: the non-existence argument for m ≤ m_c uses a Pohozaev-type multiplier on the reduced equation; the boundary term at infinity vanishes only after an integration-by-parts that assumes an extra decay on the fractional-time contribution, which needs separate justification when the profile has heavy tails.

Authors: We appreciate the referee’s careful scrutiny of the Pohozaev identity. The finite-mass condition already guarantees that both the profile and the fractional-time term (a convolution against a positive, integrable kernel) belong to L^1(ℝ^d), which is sufficient for the boundary terms at infinity to vanish after integration by parts; no additional decay beyond finite mass is required. Nevertheless, we agree that an explicit justification for the heavy-tailed regime would strengthen the exposition. We will insert a brief lemma in §5 (immediately preceding Proposition 5.2) that verifies the vanishing of the boundary term under the sole assumption of finite mass, using the explicit representation of the time-fractional operator. This addition is straightforward and keeps the argument unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; existence proof is self-contained analytical argument

full rationale

The derivation proceeds by inserting a standard self-similar ansatz (with mass-preserving scaling) into the time-fractional PME, reducing it to a stationary profile equation whose solvability is then established by fixed-point, variational, or integral-equation methods in the range m > m_c. This reduction is a methodological step, not a self-definition or renaming of a fitted quantity. The optimality claim for the range is supported by separate non-existence arguments below m_c, and the linear case m=1 is invoked only as a known explicit Fourier solution (external to the present construction). No load-bearing step reduces by construction to a prior fit, self-citation chain, or ansatz smuggled from the authors' own prior work; the central existence result therefore retains independent analytical content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no new free parameters, invented entities, or ad-hoc axioms beyond standard tools of nonlinear PDE theory; the critical exponent m_c is defined explicitly from dimension and does not require fitting.

axioms (1)

standard math Standard existence and regularity theory for fractional evolution equations and porous-medium-type nonlinearities
Implicitly invoked to justify that the self-similar ansatz yields a solvable problem in the stated parameter range.

pith-pipeline@v0.9.0 · 5429 in / 1390 out tokens · 36744 ms · 2026-05-10T17:49:28.311039+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

U(z)^m = ∫_{z}^∞ K(z,ρ) U(ρ) dρ with K(z,ρ)=ρ Q(z/ρ), Q(η)=1/Γ(1-α) ∫_1^η (1-σ^{1/b})^{-α} σ^{1-d} dσ (eqs. 9-10)
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

m_c = (d-2)_+/d; existence for m > m_c via sub/super-solutions (Theorems 1.1, 5.1, 6.1)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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