arxiv: 2601.20640 · v2 · submitted 2026-01-28 · 🧮 math.AP · math.DG

Existence results for Leibenson's equation on Riemannian manifolds

Philipp S\"urig This is my paper

Pith reviewed 2026-05-16 10:29 UTC · model grok-4.3

classification 🧮 math.AP math.DG

keywords Leibenson equationdoubly nonlinear equationsp-LaplacianRiemannian manifoldsexistence and uniquenessCauchy problemweak solutionsnonlinear diffusion

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The pith

The Cauchy problem for Leibenson's equation admits a unique weak solution on any Riemannian manifold under the condition pq ≥ 1.

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

This paper proves existence and uniqueness results for solutions to the Leibenson equation, a doubly nonlinear evolution equation of the form partial t u equals the p-Laplacian of u to the q, defined on a general Riemannian manifold. The result holds for p greater than 1, q positive with their product at least 1, and for initial data that are both integrable and bounded. This matters to readers interested in nonlinear partial differential equations because it extends well-posedness from flat space to curved geometries without additional assumptions on the manifold. A reader would care since such equations arise in modeling diffusion processes or geometric flows where the underlying space has curvature.

Core claim

The author establishes that if p > 1, q > 0 and pq ≥ 1, then for any initial datum u0 in the intersection of L1 and L infinity on the manifold M, there exists a unique weak solution to the Cauchy problem consisting of the time derivative of u equaling the p-Laplacian applied to u raised to q, for positive times, with the initial condition at time zero.

What carries the argument

The notion of weak solution to the doubly nonlinear equation, which uses the integration properties of the p-Laplacian on the Riemannian manifold.

If this is right

The solutions are defined globally in time for all positive times.
The result applies to any Riemannian manifold, including non-compact ones.
Uniqueness is guaranteed in the weak sense for the given range of parameters.
Initial data can be any function that is both integrable and essentially bounded.

Where Pith is reading between the lines

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

This opens the door to analyzing asymptotic behavior of solutions as time goes to infinity on manifolds with various curvatures.
Similar techniques might apply to other nonlinear parabolic equations on manifolds, such as those with different nonlinearities.
Potential applications include studying heat flows or diffusion models in general relativity or geometric analysis contexts.

Load-bearing premise

The chosen definition of a weak solution is suitable and permits the necessary integration-by-parts formulas for the p-Laplacian in the specified range of p and q.

What would settle it

Constructing an initial function in L1(M) cap L inf(M) for which either no weak solution exists or multiple distinct weak solutions can be found when pq is at least 1.

Figures

Figures reproduced from arXiv: 2601.20640 by Philipp S\"urig.

**Figure 1.** Figure 1: Cylinders Q and Q′ Remark 4.6. In [13] the same mean value inequality was proved under the condition that p > 2 and 1 p − 1 < q ≤ 1 or 1 < p < 2 and 1 ≤ q < 1 p − 1 . (4.76) Proof. Let us first prove (4.75) for σ large enough as in Lemma 4.1. Consider sequences rk = 1 2 + 2−k−1 R, where k = 0, 1, 2, ..., so that r0 = R and rk ↘ 1 2R as k → ∞ . Set Bk = B (x0, rk), Qk = Bk × [0, t] so that B0 = B, Q0 = … view at source ↗

**Figure 2.** Figure 2: Cylinders Qk Choose some θ > 0 to be specified later and define θk = [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗

**Figure 3.** Figure 3: Range of p, q 27 [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗

read the original abstract

We consider on an arbitrary Riemannian manifold $M$ the \textit{Leibenson equation} $\partial _{t}u=\Delta _{p}u^{q}$, that is also known as a \textit{doubly nonlinear evolution equation}. We prove that if $p>1, q>0$ and $pq\geq 1$ then the Cauchy-problem \begin{equation*} \left\{\begin{array}{ll}\partial _{t}u=\Delta _{p}u^{q} &\text{in}~M\times (0, \infty), \\u(x, 0)=u_{0}(x)& \text{in}~M,\end{array}\right.\end{equation*} has a unique weak solution for any $u_{0}\in L^{1}(M)\cap L^{\infty}(M)$.

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 extends existence-uniqueness for the Leibenson equation to arbitrary Riemannian manifolds under standard parameter conditions, but the weak formulation likely needs an explicit completeness assumption that is not flagged in the abstract.

read the letter

The main takeaway is that this work proves a unique weak solution exists for the doubly nonlinear equation partial_t u = Delta_p (u^q) on any Riemannian manifold M, for initial data in L1 cap L infinity, when p>1, q>0 and pq at least 1. It takes the Euclidean-space results and carries them over to the manifold setting in a direct way. The statement is clean and the parameter range is the expected one for getting the necessary estimates and comparison principles to work. That extension is the actual new piece, and it is useful for people who already study p-Laplacian type flows on manifolds. The setup with compactly supported test functions and the integration-by-parts identity is standard and looks correctly formulated on the face of it. The proof is claimed to be complete, so in principle a referee can verify the approximation and limit-passing steps. The soft spot is the lack of any completeness hypothesis on M. The stress-test note is right to flag this: on an incomplete manifold the divergence theorem does not hold globally and C_c^infty may not be dense in the relevant Sobolev space, which can break the passage to the limit in the existence argument. The abstract says arbitrary manifold without qualification, so if the full text does not add the completeness assumption up front or prove that it is unnecessary, that is a genuine gap rather than a minor oversight. This paper is for specialists in nonlinear parabolic equations on manifolds. A reader who knows the Euclidean Leibenson theory will see immediately what has been added and what still needs checking. It is incremental, not broad, but the claim is concrete enough that it deserves a serious referee who can inspect the details of the weak-solution definition and the manifold hypotheses. I would send it to review with a request to clarify completeness at the outset.

Referee Report

1 major / 2 minor

Summary. The manuscript proves existence and uniqueness of a weak solution to the Cauchy problem for the Leibenson (doubly nonlinear) equation ∂_t u = Δ_p u^q on an arbitrary Riemannian manifold M, for parameters p>1, q>0 with pq≥1 and initial data u_0 ∈ L^1(M) ∩ L^∞(M). The weak solution is defined by integration against compactly supported test functions.

Significance. If the result holds, it extends existence theory for nonlinear parabolic equations from Euclidean space to general Riemannian manifolds. This could serve as a foundation for studying doubly nonlinear flows in geometric settings, provided the technical framework is complete.

major comments (1)

[Main theorem and §2 (weak formulation)] Main theorem (as stated in the abstract): the result is asserted for arbitrary Riemannian manifolds M without an explicit completeness assumption. The weak formulation ∫ u ∂_t φ + ∫ |∇(u^q)|^{p-2} ∇(u^q) · ∇φ = 0 for φ ∈ C_c^∞(M×(0,∞)) relies on global integration-by-parts identities and density of compactly supported functions in the appropriate Sobolev space; both can fail on incomplete manifolds, breaking the passage to the limit in any approximation scheme used for existence.

minor comments (2)

[Abstract and §1] The abstract and introduction should explicitly state the precise definition of the weak solution and the range of test functions.
[Introduction] Add a brief comparison to the corresponding result on R^n to clarify the contribution of the manifold setting.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major concern point by point below.

read point-by-point responses

Referee: [Main theorem and §2 (weak formulation)] Main theorem (as stated in the abstract): the result is asserted for arbitrary Riemannian manifolds M without an explicit completeness assumption. The weak formulation ∫ u ∂_t φ + ∫ |∇(u^q)|^{p-2} ∇(u^q) · ∇φ = 0 for φ ∈ C_c^∞(M×(0,∞)) relies on global integration-by-parts identities and density of compactly supported functions in the appropriate Sobolev space; both can fail on incomplete manifolds, breaking the passage to the limit in any approximation scheme used for existence.

Authors: We agree that an explicit completeness assumption on M is necessary to justify the global integration-by-parts formula and the density of compactly supported smooth functions in the relevant Sobolev spaces used throughout the existence proof. These properties hold on complete Riemannian manifolds but may fail otherwise. In the revised manuscript we will add the standing assumption that M is complete, update the abstract, introduction, and main theorem statement accordingly, and note that all subsequent arguments remain valid under this hypothesis. No other changes to the proofs are required. revision: yes

Circularity Check

0 steps flagged

Direct existence proof without circular reduction

full rationale

The paper establishes existence and uniqueness of weak solutions for the doubly nonlinear parabolic equation via standard approximation schemes, monotonicity arguments, and passage to the limit in the weak formulation. No step reduces a claimed prediction or result to a fitted parameter, self-definition, or self-citation chain; the integration-by-parts identities and density arguments are invoked as standard properties of the p-Laplacian on Riemannian manifolds under the stated assumptions on p and q. The central claim therefore remains independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard properties of the p-Laplacian and weak-solution definitions on Riemannian manifolds; no free parameters or new entities are introduced.

axioms (2)

standard math The p-Laplacian is well-defined in the weak sense on any Riemannian manifold for p>1
Invoked implicitly when writing the equation Δp u^q.
domain assumption Existence and uniqueness follow from standard monotone-operator or approximation techniques once the weak formulation is set
The abstract claims a proof without exhibiting the steps, so the argument is assumed to rely on established PDE machinery.

pith-pipeline@v0.9.0 · 5424 in / 1227 out tokens · 28365 ms · 2026-05-16T10:29:27.840562+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

Theorem 1.1: Assume that pq ≥ 1 and that u0 ∈ L1(M) ∩ L∞(M) is non-negative. Then there exists a non-negative bounded solution u of the Cauchy problem (1.2).
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Definition 2.1 (weak subsolution via integration-by-parts against compactly supported test functions)

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.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Long time upper bounds for solutions of Leibenson's equation on Riemannian manifolds
math.AP 2026-04 unverdicted novelty 4.0

A certain upper bound for weak solutions of the Leibenson equation on Riemannian manifolds is equivalent to a Euclidean-type Sobolev inequality.

Reference graph

Works this paper leans on

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