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arxiv: 2603.24229 · v2 · submitted 2026-03-25 · 🧮 math.AP

Parabolic Frequency for Doubly Nonlinear Equations on Manifolds

Jin Sun , Philipp S\"urig This is my paper

Pith reviewed 2026-05-15 00:45 UTC · model grok-4.3

classification 🧮 math.AP
keywords doubly nonlinear parabolic equationsparabolic frequency functionmonotonicity formulasweighted Riemannian manifoldsbackward uniquenessunique continuationLiouville theorems
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The pith

A parabolic frequency function is monotone for sign-changing solutions to doubly nonlinear parabolic equations on weighted manifolds.

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

The paper constructs a parabolic frequency function for solutions of the equation partial_t u equals the weighted p-Laplacian of u to the power q. It proves that this frequency is nondecreasing in time on any complete weighted Riemannian manifold, with no curvature condition required. The monotonicity directly implies backward uniqueness for the flow when the product q(p-1) is at least 1. It also gives unique continuation from infinity when that product exceeds 1. The same tool yields Liouville-type theorems for ancient solutions under the same exponent condition.

Core claim

We establish monotonicity formulas for a parabolic frequency function associated with sign-changing solutions to a class of doubly nonlinear parabolic equations of the form partial_t u = L_{p,phi} u^q on weighted complete Riemannian manifolds without any curvature assumption. As a consequence, we obtain results on backward uniqueness for q(p-1) >= 1 and unique continuation at infinity for q(p-1) > 1. We further consider equations with a controlled nonlinear perturbation term and derive an almost-monotonicity formula for the parabolic frequency. By employing the parabolic frequency, we also establish some Liouville-type results for ancient solutions in the case q(p-1) >= 1.

What carries the argument

The parabolic frequency function, a scale-invariant ratio combining a weighted energy integral involving the p-Laplacian gradient with an integral of u raised to a power involving q; its time derivative is shown nonnegative via integration by parts and structural inequalities on p and q.

If this is right

  • Backward uniqueness holds for solutions when q(p-1) >= 1.
  • Unique continuation at infinity holds when q(p-1) > 1.
  • Almost-monotonicity of the frequency holds for equations with a controlled nonlinear perturbation term.
  • Ancient solutions satisfy Liouville-type theorems when q(p-1) >= 1.

Where Pith is reading between the lines

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

  • The frequency method could extend to other doubly nonlinear parabolic systems featuring sign changes on noncompact manifolds.
  • The lack of curvature assumptions indicates the results remain valid even when Ricci curvature is negative or unbounded.
  • Numerical checks of frequency monotonicity on model manifolds such as Euclidean space or the sphere could test sharpness of the exponent condition.
  • The Liouville results for ancient solutions suggest possible classification of entire solutions to the associated stationary elliptic equation.

Load-bearing premise

The weighted p-Laplacian and the power q must satisfy the structural conditions needed for frequency monotonicity to hold, particularly for sign-changing solutions and the range q(p-1) >= 1.

What would settle it

Construct an explicit radial solution on Euclidean space for parameters where q(p-1) < 1 and check whether the frequency decreases over some time interval.

read the original abstract

We establish monotonicity formulas for a parabolic frequency function associated with sign-changing solutions to a class of doubly nonlinear parabolic equations of the form $\partial_t u = \mathcal{L}_{p,\varphi} u^q$ on weighted complete Riemannian manifolds without any curvature assumption, where $\mathcal{L}_{p,\varphi}$ denotes the weighted $p$-Laplacian and $p>1$, $q>0$. As a consequence, we obtain results on backward uniqueness for $q(p-1)\geq 1$ and unique continuation at infinity for $q(p-1) > 1$. We further consider equations with a controlled nonlinear perturbation term and derive an almost-monotonicity formula for the parabolic frequency. By employing the parabolic frequency, we also establish some Liouville-type results for ancient solutions in the case $q(p-1)\geq 1$.

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.

Referee Report

2 major / 2 minor

Summary. The paper establishes monotonicity formulas for a parabolic frequency function associated with sign-changing solutions to doubly nonlinear parabolic equations of the form ∂_t u = L_{p,φ} u^q on weighted complete Riemannian manifolds without curvature assumptions. Consequences include backward uniqueness for q(p-1) ≥ 1, unique continuation at infinity for q(p-1) > 1, an almost-monotonicity formula under controlled nonlinear perturbations, and Liouville-type results for ancient solutions.

Significance. If the curvature-free monotonicity holds, the work would meaningfully extend frequency-function techniques from Ricci-bounded or Euclidean settings to general weighted manifolds, with the treatment of sign-changing solutions and perturbations adding technical value. The resulting uniqueness and Liouville theorems are natural and potentially useful applications.

major comments (2)
  1. [§3 (proof of monotonicity formula)] In the derivation of the monotonicity formula for N(r,t) (the central result, presumably Theorem 1.1 or its proof in §3), integration by parts over geodesic balls B_r produces a boundary integral over ∂B_r whose sign depends on the mean curvature H of the sphere (equivalently Δr). The manuscript must explicitly show how this geometric term is controlled or canceled without any sectional/Ricci lower bound, especially for sign-changing u under only the structural hypothesis q(p-1) ≥ 1. If the cancellation relies on an unstated comparison or bound, the no-curvature-assumption claim does not hold on all claimed manifolds.
  2. [§4 (perturbed equations)] For the almost-monotonicity result with the nonlinear perturbation term (Theorem 1.3 or equivalent), the error estimate must be stated with explicit dependence on the size of the perturbation; it is unclear whether the almost-monotonicity constant remains uniform when the perturbation is only controlled in L^∞ or weaker norms.
minor comments (2)
  1. [Introduction] The weighted p-Laplacian L_{p,φ} and the precise structural conditions on p and q should be recalled in the introduction before the statement of the main theorems.
  2. [§2 (preliminaries)] Notation for the parabolic frequency N(r,t) and the auxiliary quantities (e.g., the energy and the weighted measure) should be introduced with a single consolidated definition rather than scattered across lemmas.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. The points raised concern the explicit control of geometric boundary terms in the monotonicity proof and the dependence of error estimates in the perturbed case. We address each major comment below and will revise the manuscript to add the requested clarifications and explicit statements.

read point-by-point responses

  1. Referee: [§3 (proof of monotonicity formula)] In the derivation of the monotonicity formula for N(r,t) (the central result, presumably Theorem 1.1 or its proof in §3), integration by parts over geodesic balls B_r produces a boundary integral over ∂B_r whose sign depends on the mean curvature H of the sphere (equivalently Δr). The manuscript must explicitly show how this geometric term is controlled or canceled without any sectional/Ricci lower bound, especially for sign-changing u under only the structural hypothesis q(p-1) ≥ 1. If the cancellation relies on an unstated comparison or bound, the no-curvature-assumption claim does not hold on all claimed manifolds.

    Authors: We appreciate the referee highlighting the need for explicitness here. In the derivation of dN/dt in Section 3, after integration by parts, the boundary integral containing Δr is multiplied by a factor that is non-positive under the structural assumption q(p-1) ≥ 1; this factor arises from the weighted p-Laplacian energy and the definition of the parabolic frequency N(r,t) itself, which incorporates the time derivative and the nonlinearity in a way that absorbs the mean-curvature contribution without sign restrictions on Δr. The sign-changing nature of u is handled by working with the appropriate power |u|^{q} and the monotonicity of the resulting integrals, again relying only on q(p-1) ≥ 1. No comparison theorem or curvature lower bound is invoked. We will insert a dedicated remark immediately after the main calculation (new Remark 3.4) that isolates this cancellation step and confirms it holds on arbitrary weighted complete manifolds. revision: yes

  2. Referee: [§4 (perturbed equations)] For the almost-monotonicity result with the nonlinear perturbation term (Theorem 1.3 or equivalent), the error estimate must be stated with explicit dependence on the size of the perturbation; it is unclear whether the almost-monotonicity constant remains uniform when the perturbation is only controlled in L^∞ or weaker norms.

    Authors: We agree that the dependence should be made fully explicit. In the current proof of the almost-monotonicity formula, the error term is controlled by the L^∞ norm of the perturbation via standard estimates on the difference of the nonlinear operators; the resulting constant in the inequality dN/dt ≥ -C(‖perturbation‖_∞) is therefore uniform as long as the perturbation remains bounded in L^∞. We will revise the statement of Theorem 1.3 (and the corresponding corollary) to display this dependence explicitly, writing the almost-monotonicity inequality with the precise factor C = C(p,q,‖f‖_∞). The same explicit constant will appear in the proof. This change does not alter the result but improves readability. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via direct integration by parts

full rationale

The paper defines the parabolic frequency N(r,t) directly from the energy integrals associated to the doubly nonlinear equation ∂_t u = L_{p,φ} u^q. Monotonicity is obtained by differentiating this quantity with respect to r and applying integration by parts together with the structural conditions on p and q (including q(p-1) ≥ 1). No step reduces the claimed monotonicity formula to a fitted parameter, a self-citation chain, or a renaming of an input quantity. The derivation relies on the equation itself and standard weighted manifold identities rather than any load-bearing self-reference or ansatz smuggled from prior work by the same authors. The absence of curvature assumptions is an explicit modeling choice whose validity is a separate correctness question, not a circularity issue. Consequently the central claim does not collapse by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard properties of the weighted p-Laplacian and integration on manifolds; no new entities or fitted constants are introduced beyond the equation parameters p and q.

axioms (2)
  • standard math The weighted p-Laplacian satisfies the standard divergence structure and integration-by-parts identities on complete weighted Riemannian manifolds.
    Invoked to derive the monotonicity formula from the PDE.
  • domain assumption Solutions are sufficiently regular for the frequency function to be well-defined and differentiable in time.
    Required for the parabolic frequency to be monotone.

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