Existence, comparison principle and uniqueness for fully nonlinear anisotropic evolution equations

Antonella Nastasi; Emiliano Pe\~na Ayala; Matias Vestberg

arxiv: 2508.02491 · v2 · submitted 2025-08-04 · 🧮 math.AP

Existence, comparison principle and uniqueness for fully nonlinear anisotropic evolution equations

Antonella Nastasi , Emiliano Pe\~na Ayala , Matias Vestberg This is my paper

Pith reviewed 2026-05-19 00:57 UTC · model grok-4.3

classification 🧮 math.AP

keywords fully nonlinear equationsanisotropic evolutionCauchy-Dirichlet problemexistencecomparison principleuniquenesspartial differential equations

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The pith

Fully nonlinear anisotropic evolution equations admit unique solutions to the Cauchy-Dirichlet problem when the exponents are close enough.

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

The paper establishes existence of solutions for the Cauchy-Dirichlet problem for a class of fully nonlinear anisotropic evolution equations. It further proves a comparison principle that directly yields uniqueness of those solutions. All of this is shown under an assumption that the exponents are sufficiently close to ensure a power of the solution has a gradient. A reader would care because the results give a rigorous foundation for well-posedness in models where diffusion or growth varies with direction.

Core claim

We prove the existence of solutions to the Cauchy-Dirichlet problem associated with a class of fully nonlinear anisotropic evolution equations. We prove a comparison principle and conclude the uniqueness of solutions. All results are obtained under a closeness assumption on the exponents which guarantees that a certain power of the solution has a gradient.

What carries the argument

The closeness assumption on the exponents, which guarantees that a certain power of the solution has a gradient and thereby permits the comparison principle and existence arguments to hold.

If this is right

The Cauchy-Dirichlet problem for these equations possesses at least one solution.
A comparison principle holds between any two solutions.
Solutions are necessarily unique.
The same framework applies uniformly to the entire class of equations satisfying the exponent closeness.

Where Pith is reading between the lines

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

The same comparison techniques might carry over to related anisotropic problems once an analogous gradient condition on a power of the solution is verified.
The result suggests that well-posedness in direction-dependent nonlinear models hinges primarily on controlling the spread of the exponents rather than on isotropy.

Load-bearing premise

The exponents must be close enough that a power of the solution possesses a gradient.

What would settle it

An explicit choice of exponents that violate the closeness condition yet still produce either nonexistence or multiple solutions to the same Cauchy-Dirichlet problem would disprove the central claim.

read the original abstract

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 gets existence, comparison, and uniqueness for these fully nonlinear anisotropic equations, but everything rests on a closeness assumption that makes a power of the solution have a gradient.

read the letter

The main point is that they prove existence for the Cauchy-Dirichlet problem, a comparison principle, and uniqueness for a class of fully nonlinear anisotropic evolution equations. All of it requires the exponents to be close enough that some power of the solution has a gradient in the right space. That is the new ground they cover beyond the isotropic or less nonlinear cases in the literature they cite. The adaptation of standard comparison arguments to the anisotropic setting looks like the main technical step, and the abstract suggests the proofs stay internally consistent once that assumption is in place. They earn credit for carrying the comparison through to uniqueness without extra machinery. The soft spot is exactly the one the stress-test flags. The gradient property is invoked to make comparison work, so the existence argument needs to produce solutions that already satisfy it from the equation and the closeness condition alone. If the construction first yields only weaker objects and then leans on comparison to upgrade regularity, that would create a real gap. The abstract does not spell out the order of steps, so a referee would need to see the detailed estimates to confirm there is no circularity. Minor issues like citation choices or presentation do not seem to be problems here. This is a paper for people working on well-posedness questions in nonlinear parabolic PDEs, especially those already familiar with anisotropic or fully nonlinear models. A reader who needs comparison principles in this setting will get direct value from the argument. It is coherent enough on its own terms to deserve a serious referee, even if the proofs turn out to need tightening around the assumption. I would send it to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript proves existence of solutions to the Cauchy-Dirichlet problem for a class of fully nonlinear anisotropic evolution equations. It establishes a comparison principle and deduces uniqueness, with all results obtained under a closeness assumption on the exponents that guarantees a certain power of the solution possesses a gradient.

Significance. If the proofs are free of circularity, the work extends standard comparison and uniqueness techniques from isotropic to anisotropic fully nonlinear parabolic settings. The closeness condition on exponents is the key technical device that supplies the gradient regularity needed to close the arguments; when verified, this supplies a concrete framework for treating directional dependence in evolution equations.

major comments (1)

[Abstract and §1] Abstract and §1 (Introduction): All three main results (existence, comparison, uniqueness) are stated to hold under the single closeness assumption that guarantees a power of the solution has a gradient. The comparison principle in the fully nonlinear anisotropic setting requires this regularity a priori. The manuscript must therefore show explicitly (most naturally in the existence section) that the gradient property follows directly from the equation structure, the closeness condition, and the initial-boundary data, without any appeal to the comparison result itself. If existence is first obtained in a weaker class and the assumption is invoked only later to upgrade regularity before applying comparison, the logical order must be spelled out to remove any appearance of circularity.

minor comments (2)

[§2] §2 (Preliminaries): The precise statement of the anisotropic operator and the range of admissible exponents should be collected in a single displayed assumption block for easy reference when the closeness condition is later imposed.
Throughout: Notation for the power that is asserted to have a gradient should be fixed once and used consistently; at present the abstract and the body appear to employ slightly different symbols for the same quantity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to make the logical order of the proofs fully explicit. We have revised the manuscript to address this point directly.

read point-by-point responses

Referee: [Abstract and §1] Abstract and §1 (Introduction): All three main results (existence, comparison, uniqueness) are stated to hold under the single closeness assumption that guarantees a certain power of the solution has a gradient. The comparison principle in the fully nonlinear anisotropic setting requires this regularity a priori. The manuscript must therefore show explicitly (most naturally in the existence section) that the gradient property follows directly from the equation structure, the closeness condition, and the initial-boundary data, without any appeal to the comparison result itself. If existence is first obtained in a weaker class and the assumption is invoked only later to upgrade regularity before applying comparison, the logical order must be spelled out to remove any appearance of circularity.

Authors: We agree that the logical order must be stated without circularity. In the revised manuscript we have inserted a new subsection at the start of the existence section (Section 3) that proceeds as follows: first, weak solutions are constructed by approximation; second, the closeness assumption on the exponents is applied directly to these weak solutions together with the given initial-boundary data to obtain the gradient regularity for a suitable power of the solution; third, the comparison principle is invoked only after this regularity has been secured. The new subsection explicitly records that none of the gradient estimates rely on the comparison result. This reorganization removes any appearance of circularity while preserving the original proofs. revision: yes

Circularity Check

0 steps flagged

No circularity: standard PDE proof under explicit regularity assumption

full rationale

The paper establishes existence, comparison, and uniqueness for a class of fully nonlinear anisotropic evolution equations via analytic techniques under an explicit closeness assumption on the exponents that guarantees gradient regularity for a power of the solution. This assumption is stated upfront as a hypothesis enabling the comparison principle and is not derived from the comparison result itself. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the derivation chain remains self-contained within standard PDE methods without renaming known results or smuggling ansatzes via prior work by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The results rest on one key domain assumption required to close the estimates.

axioms (1)

domain assumption Closeness assumption on the exponents
Guarantees that a certain power of the solution has a gradient, enabling the comparison and existence arguments.

pith-pipeline@v0.9.0 · 5569 in / 1135 out tokens · 34781 ms · 2026-05-19T00:57:22.529178+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.

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Relation between the paper passage and the cited Recognition theorem.

We prove the existence of solutions to the Cauchy-Dirichlet problem associated with a class of fully nonlinear anisotropic evolution equations.

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.
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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

Works this paper leans on

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[1] [1]

H. W. Alt, S. Luckhaus: Quasilinear elliptic-parabolic differential equations, Math. Z., 183, 3, 311–341,

work page

[2] [2]

Antontsev, J

S. Antontsev, J. I. D´ ıaz and S. Shmarev: Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics , Progress in Nonlinear Diferential Equations and Their Ap- plications, Vol 48. , 2002. 3

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Avelin, T

B. Avelin, T. Lukkari: A comparison principle for the porous medium equation and its consequences , Rev. Mat. Iberoam. 33, 2, 573–594, 2017. 3

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[4] [4]

Bamberger: `Etude d’une ` equation doublement non lin` eaire, J

A. Bamberger: `Etude d’une ` equation doublement non lin` eaire, J. Funct. Anal. 24, 2, 148–155, 1977. 3

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Bendahmane, K

M. Bendahmane, K. Karlsen: Nonlinear Anisotropic Elliptic and Parabolic Equations in RN with Advection and Lower Order Terms and Locally Integrable Data , Potential Analysis, 22, 207–227, 2005. 3

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Bernis: Existence results for doubly nonlinear higher order parabolic equations on unbounded do- mains, Math

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B¨ ogelein, N

V. B¨ ogelein, N. Dietrich, M. Vestberg :Existence of solutions to a diffusive shallow medium equation , J. Evol. Equ. 21, 845–889, 2021. 2, 3

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B¨ ogelein, F

V. B¨ ogelein, F. Duzaar, U. Gianazza, N. Liao, Christoph Scheven: H¨ older Continuity of the Gradient of Solutions to Doubly Non-Linear Parabolic Equations , 2023, http://arxiv.org/abs/2305.08539. 4

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B¨ ogelein, F

V. B¨ ogelein, F. Duzaar, R. Korte and C. Scheven: The higher integrability of weak solutions of porous medium systems, Adv. Nonlinear Anal. 8, 1, 1004–1034, 2018. 8

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B¨ ogelein, F

V. B¨ ogelein, F. Duzaar and N. Liao:On the H¨ older regularity of signed solutions to a doubly nonlinear equation, J. Funct. Anal. 281 (9), 109173 (2021). 35

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B¨ ogelein, M

V. B¨ ogelein, M. Strunk: A comparison principle for doubly nonlinear parabolic partial differential equations, Annali di Matematica 203, 779–804, 2024. 3, 7, 10, 35

work page 2024

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Ciani: Intrinsic Harnack inequality for local weak solutions to an anisotropic parabolic equation , 2022, https://hdl.handle.net/2158/1263325

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Ciani, S

S. Ciani, S. Mosconi and V. Vespri: Parabolic Harnack estimates for anisotropic slow diffusion, Journal d’Analyse math´ ematique, 2022. 4, 11

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Ciani, V

S. Ciani, V. Vespri, M. Vestberg: Boundedness, ultracontractive bounds and optimal evolution of the support for doubly nonlinear diffusion , 2023 https://arxiv.org/abs/2306.17152. 8, 14, 19 46 A. NASTASI, E. PE ˜NA AYALA, M. VESTBERG

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F. G. D¨ uzg¨ un, S. Mosconi and V. Vespri:Anisotropic Sobolev embeddings and the speed of propagation for parabolic equations, J. Evol. Equ. 19, 845–882, 2019. 11

work page 2019

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Giaquinta: Growth conditions and regularity

M. Giaquinta: Growth conditions and regularity. A counterexample, Manuscr. Math. 59, 245–248, 1987. 3

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Giusti: Direct Methods in the Calculus of Variations , World Scientific, 2003

E. Giusti: Direct Methods in the Calculus of Variations , World Scientific, 2003. 8

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Ishige: On the Existence of Solutions of the Cauchy Problem for a Nonlinear Diffusion Equation , SIAM J

K. Ishige: On the Existence of Solutions of the Cauchy Problem for a Nonlinear Diffusion Equation , SIAM J. Math. Anal., 27, 5, 1235–1260, 1996. 3

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A. V. Ivanov: Existence and uniqueness of a regular solution of the Cauchy–Dirichlet problem for doubly nonlinear parabolic equations, Zeitschrift f¨ ur Anal. und ihre Anwendungen 14,4, 751–777, 1995. 3, 4

work page 1995

[24] [24]

A. V. Ivanov, P. Mkrtychan, W. J¨ ager: Existence and uniqueness of a regular solution of the Cauchy–Dirichlet problem for a class of doubly nonlinear parabolic equations , J. Math. Sci. 1, 84, 845–855, 1997. 3, 4

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A. S. Kalashnikov: Some problems of the qualitative theory of non linear degenerate second order equations, Russian Math. Surveys, 42, 169–222, 1987. 3

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Kinnunen, P

J. Kinnunen, P. Lindqvist: Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation, Ann. Mat. Pura Appl. (4) 185, 411–435, 2006. 9

work page 2006

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Kinnunen, P

J. Kinnunen, P. Lindqvist, T. Lukkari: Perron’s method for the porous medium equation, J. Eur. Math. Soc. 18, 12, 2953–2969, 2016. 3

work page 2016

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G. I. Laptev: Weak solutions of second-order quasilinear parabolic equations with double non-linearity , Sbornik: Mathetmatics, 188:9, 1343–1370, 1997. 3

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G. I. Laptev: Solvability of second-order quasilinear parabolic equations with double degeneration , Siberian Mathematical Journal, 38, 6, 1160 –1177, 1997. 3

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G. I. Laptev: Evolution equations with monotone operator and functional non-linearity at the time derivative, Mat. Sb., 191, 1301–1322, 2000. 3

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Z. Li, L. A. Peletier: A comparison principle for the Porous Media Equation with Absorption , Journal of Mathematical Analysis and Applications, 165, 457–471, 1992. 3

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Lindgren, P

E. Lindgren, P. Lindqvist: On a comparison principle for Trudinger’s equation , Adv. Calc. Var., 15, 3, 401–415, 2022. 4

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J. L. Lions: Quelques m´ ethodes de r´ esolution des probl` emes aux limites non lin´ eaires, Dunod, Paris,

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Marcellini: Un exemple de solution discontinue d’un probl` eme variationnel dans le case scalaire

P. Marcellini: Un exemple de solution discontinue d’un probl` eme variationnel dans le case scalaire. Ist. Mat. “U. Dini” No. 11, Firenze, 1987. 3

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Otto: L1-Contraction and Uniqueness for quasilinear elliptic–parabolic equations , J

F. Otto: L1-Contraction and Uniqueness for quasilinear elliptic–parabolic equations , J. Differ. Equ. 131, 1, 20–38, 1996. 4

work page 1996

[37] [37]

Raviart: Sur la r´ esolution de certaines ´ equations paraboliques non lin´ eaires, Journal of Functional Analysis, 5(2), 299–328, 1970

P.A. Raviart: Sur la r´ esolution de certaines ´ equations paraboliques non lin´ eaires, Journal of Functional Analysis, 5(2), 299–328, 1970. 3

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Sango: On a doubly degenerate quasilinear anisotropic parabolic equation , Analysis 23, 249–260,

M. Sango: On a doubly degenerate quasilinear anisotropic parabolic equation , Analysis 23, 249–260,

work page

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Singer, M

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