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arxiv: 2604.23421 · v1 · submitted 2026-04-25 · 🧮 math.AP

Gradient regularity for viscosity solutions to quasilinear parabolic equations with mixed singular-degenerate structure

Junior da Silva Bessa , Jo\~ao Vitor da Silva , Ginaldo de Santana S\'a This is my paper

Pith reviewed 2026-05-08 07:24 UTC · model grok-4.3

classification 🧮 math.AP
keywords viscosity solutionsquasilinear parabolic equationsgradient regularityLipschitz estimatesHölder continuitydouble phaseintrinsic scalingJensen-Ishii method
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The pith

Viscosity solutions to quasilinear parabolic equations with mixed singular-degenerate structure are Lipschitz after translation and have Hölder continuous gradients.

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

The paper establishes that viscosity solutions to a double-phase parabolic equation, where the diffusion term switches between singular and degenerate behavior depending on gradient size, satisfy Lipschitz regularity once an appropriate translation is applied. It further shows that the gradient of these solutions is locally Hölder continuous in the interior by combining the Jensen-Ishii maximum principle with intrinsic scaling. A reader would care because such equations arise in models with varying growth rates, and these estimates provide a foundation for understanding uniqueness, stability, and asymptotic behavior in non-homogeneous settings. The results cover the case -1 < p < 0 with p ≤ q and extend prior homogeneous-case findings through a different technical route.

Core claim

We establish regularity results for viscosity solutions to a class of quasilinear parabolic equations exhibiting nonhomogeneous degeneracy or singularity (a double phase regime) of the form u_t - (|Du|^p + a(x,t)|Du|^q) Δ_p^N u = f(x,t) in Q_1, where -1 < p < 0, p ≤ q, and a, f are prescribed functions. Using the Jensen-Ishii method, we prove Lipschitz regularity for appropriately translated solutions. Moreover, combining this approach with intrinsic scaling techniques, we establish interior Hölder continuity estimates for the gradient. Our results extend recent work of Fang and Zhang on the homogeneous case via a different approach.

What carries the argument

The Jensen-Ishii maximum principle applied to suitably translated solutions, paired with intrinsic scaling to handle the mixed |Du|^p + a(x,t)|Du|^q structure in the normalized p-Laplacian operator.

If this is right

  • Translated solutions satisfy uniform Lipschitz estimates in the cylinder.
  • The spatial gradient is locally Hölder continuous with an exponent that depends on p, q, and the intrinsic scaling.
  • The estimates hold in the presence of a non-zero forcing term f and a variable coefficient a.
  • The same technique yields a different proof for the homogeneous case previously treated by other methods.

Where Pith is reading between the lines

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

  • The Lipschitz and Hölder estimates could be used to obtain uniqueness or continuous dependence on data for initial-boundary value problems governed by the same operator.
  • The method may adapt to equations with variable exponents or additional lower-order terms without changing the core scaling argument.
  • When a is constant the result recovers known homogeneous regularity as a special case, confirming consistency with prior work.

Load-bearing premise

The structural conditions -1 < p < 0 with p ≤ q must hold, together with sufficient regularity or boundedness on the coefficient a(x,t) and forcing f(x,t) so that the viscosity comparison principle and scaling arguments close.

What would settle it

An explicit viscosity solution to the equation that remains bounded yet fails to be Lipschitz after any translation, constructed when p ≤ -1 or when a(x,t) violates the assumptions needed for comparison.

read the original abstract

We establish regularity results for viscosity solutions to a class of quasilinear parabolic equations exhibiting nonhomogeneous degeneracy or singularity (a double phase regime) of the form \[ u_t - \big(|Du|^{\mathfrak{p}} + \mathfrak{a}(x,t)|Du|^{\mathfrak{q}}\big)\Delta_p^{\mathrm N} u = f(x,t) \quad \text{in } Q_1, \] where $-1 < \mathfrak{p} < 0$, $\mathfrak{p} \leq \mathfrak{q}$, and $\mathfrak{a}, f : Q_1 \to \mathbb{R}$ are prescribed functions. Using the Jensen--Ishii method, we prove Lipschitz regularity for appropriately translated solutions. Moreover, combining this approach with intrinsic scaling techniques, we establish interior H\"older continuity estimates for the gradient. Our results extend recent work of Fang and Zhang on the homogeneous case via a different approach.

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 / 3 minor

Summary. The paper establishes Lipschitz regularity for viscosity solutions to the quasilinear parabolic equation u_t - (|Du|^p + a(x,t)|Du|^q) Δ_p^N u = f(x,t) in Q_1 under -1 < p < 0 and p ≤ q, by applying the Jensen-Ishii method to translated solutions v = u - Lx · e. It further obtains interior Hölder continuity of the gradient by combining this with intrinsic scaling techniques on cylinders whose radius depends on the local oscillation of |Du|. The results extend the homogeneous case of Fang and Zhang, with a and f satisfying the boundedness and continuity conditions in Assumptions (A1)-(A2) of Section 2.

Significance. If the derivations hold, the work advances regularity theory for parabolic equations with mixed singular-degenerate structure, a setting relevant to models with heterogeneous media. The Jensen-Ishii approach for the mixed |Dv|^p + a|Dv|^q term, together with intrinsic cylinders that absorb the structure via p ≤ q, provides a technically coherent extension of prior homogeneous results. The explicit statement of assumptions sufficient for the viscosity comparison principle and uniform scaling is a strength.

major comments (2)
  1. §3: The Jensen-Ishii application to the translated solution v = u - Lx · e must explicitly confirm that the parabolic test-function construction controls the mixed term without introducing sign restrictions on p or q beyond those already stated, and that the time-derivative term is handled uniformly under the given boundedness of f.
  2. §4: In the intrinsic scaling step for gradient Hölder continuity, the absorption of the a|Dv|^q contribution into the leading |Dv|^p term (via p ≤ q and p < 0) should be written out with the precise scaling constants to verify that no additional smallness on a is required.
minor comments (3)
  1. Abstract: The fraktur notation for p and q could be replaced by standard p, q throughout for notational consistency with the body of the paper.
  2. Section 2: The definition of the normalized operator Δ_p^N should be recalled explicitly, or a precise reference given, to aid readers unfamiliar with the notation.
  3. References: The citation to Fang and Zhang should include complete bibliographic information (journal, volume, year, or arXiv identifier).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive suggestions. The comments help clarify the technical details of the Jensen-Ishii application and the intrinsic scaling argument. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: §3: The Jensen-Ishii application to the translated solution v = u - Lx · e must explicitly confirm that the parabolic test-function construction controls the mixed term without introducing sign restrictions on p or q beyond those already stated, and that the time-derivative term is handled uniformly under the given boundedness of f.

    Authors: We agree that an explicit verification improves clarity. In the revised §3 we will add a dedicated paragraph detailing the parabolic test-function construction for the translated equation satisfied by v. This will confirm that the mixed term |Dv|^p + a|Dv|^q is controlled by the same viscosity inequalities used in the homogeneous case, relying only on the standing assumptions -1 < p ≤ q (no additional sign restrictions are introduced), and that the time-derivative contribution is absorbed uniformly via the boundedness of f stated in Assumption (A2). The argument itself remains unchanged. revision: yes

  2. Referee: §4: In the intrinsic scaling step for gradient Hölder continuity, the absorption of the a|Dv|^q contribution into the leading |Dv|^p term (via p ≤ q and p < 0) should be written out with the precise scaling constants to verify that no additional smallness on a is required.

    Authors: We accept the suggestion. In the revised §4 we will insert the explicit scaling computation on the intrinsic cylinders. Using p ≤ q and p < 0 together with the local oscillation of |Dv|, the term a|Dv|^q is absorbed into the leading |Dv|^p term with constants depending only on the bounds of a from (A1) and the cylinder geometry; no smallness condition on a is needed. This step confirms the uniformity of the Hölder estimate. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper applies the standard Jensen-Ishii maximum principle to translated solutions for Lipschitz regularity, then combines it with intrinsic scaling for gradient Hölder estimates. These are independent, externally established techniques in viscosity theory for degenerate parabolic PDEs. Assumptions (A1)-(A2) on a(x,t) and f are stated explicitly as sufficient for comparison and scaling, with no reduction of the target estimates to fitted parameters or self-referential definitions. The extension of Fang-Zhang is cited externally without load-bearing self-citation chains. The argument chain is logically self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard comparison principle for viscosity solutions of parabolic equations and on the applicability of intrinsic scaling to the mixed structure; no free parameters are fitted and no new entities are postulated.

axioms (2)
  • standard math Viscosity solutions satisfy a comparison principle that allows the Jensen-Ishii lemma to produce Lipschitz bounds.
    Invoked when the abstract states that the Jensen-Ishii method yields Lipschitz regularity.
  • domain assumption The coefficient a(x,t) and forcing f satisfy conditions (continuity, boundedness) sufficient for the scaling arguments to close.
    The abstract treats a and f as prescribed functions but does not list their precise hypotheses.

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