Bounded solutions and interpolative gap bounds for degenerate parabolic double phase problems

Bogi Kim; Jehan Oh

arxiv: 2511.13454 · v2 · submitted 2025-11-17 · 🧮 math.AP

Bounded solutions and interpolative gap bounds for degenerate parabolic double phase problems

Bogi Kim , Jehan Oh This is my paper

Pith reviewed 2026-05-17 20:45 UTC · model grok-4.3

classification 🧮 math.AP

keywords double phase problemsparabolic equationshigher integrabilitydegenerate equationsgap conditiongradient estimatesHölder coefficients

0 comments

The pith

Gradient higher integrability holds for solutions to parabolic double phase equations under the gap condition q ≤ p + α when the solution is bounded.

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

The paper establishes that weak solutions to the degenerate parabolic double phase equation achieve higher integrability of the gradient when the gap between the exponents p and q is controlled by the Hölder regularity α of the coefficient a. For bounded solutions this holds as long as q ≤ p + α. For solutions that lie in C(0,T; L^s(Ω)) with s ≥ 2 the allowable gap shrinks to q ≤ p + sα/(n + s). These conditions interpolate between earlier gap bounds known for the parabolic double phase setting and rely on the precise divergence structure of the operator together with the Hölder continuity of a.

Core claim

Weak solutions to u_t − div(|Du|^{p−2}Du + a(x,t)|Du|^{q−2}Du) = 0 with a ∈ C^{α,α/2}(Ω_T) possess gradient higher integrability whenever the gap condition q ≤ p + α is satisfied for bounded solutions, or q ≤ p + sα/(n + s) is satisfied when u belongs to C(0,T; L^s(Ω)) for s ≥ 2.

What carries the argument

The double-phase divergence operator |Du|^{p−2}Du + a(x,t)|Du|^{q−2}Du whose oscillation is controlled by the Hölder continuity of a, combined with Gehring-type higher-integrability arguments adapted to the parabolic setting.

If this is right

Bounded solutions gain |Du| ∈ L^{p+δ} locally for some δ > 0 under the gap q ≤ p + α.
Solutions with extra time integrability u ∈ C(0,T; L^s) gain higher integrability under the reduced gap q ≤ p + sα/(n + s).
The results interpolate between the gap bounds previously obtained for the parabolic double-phase problem.
The Hölder modulus of a directly determines the size of the allowable gap between p and q.

Where Pith is reading between the lines

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

The same technique may extend to systems or to equations with lower-order terms once the precise double-phase structure is preserved.
Numerical tests could check whether the gap thresholds are sharp by constructing explicit solutions that lose integrability exactly when q surpasses the stated bound.
The interpolation suggests that higher time integrability of u can partially compensate for a larger spatial gap between p and q.

Load-bearing premise

The coefficient a must be Hölder continuous of order α in space and α/2 in time, and the equation must have exactly this double-phase divergence form.

What would settle it

A bounded weak solution for which q exceeds p + α yet the gradient fails to gain any extra integrability.

read the original abstract

We establish gradient higher integrability results for weak solutions to degenerate parabolic equations of double phase type $$ u_t-\operatorname{div} \left(|Du|^{p-2}Du + a(x,t)|Du|^{q-2}Du\right)=0 $$ in $\Omega_T := \Omega\times (0,T)$, where $a(\cdot)\in C^{\alpha,\frac{\alpha}{2}}(\Omega_T)$. For bounded solutions, we prove that the result holds under the gap condition $$ q \leq p + \alpha. $$ Moreover, for solutions with $$ u\in C(0,T;L^s(\Omega)), \quad s \geq 2, $$ we obtain higher integrability under the gap condition $$ q \leq p + \frac{s\alpha}{n+s}. $$ These results provide an interpolation between the gap bounds in the parabolic double phase setting.

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's main advance is an explicit interpolation of the gap bounds for gradient higher integrability in the parabolic double-phase setting.

read the letter

The main takeaway is that the authors interpolate the allowable gap between p and q for gradient higher integrability in degenerate parabolic double-phase equations. They get the full q ≤ p + α when solutions are bounded, but relax it to q ≤ p + sα/(n+s) once you assume u is continuous in time with values in L^s for s ≥ 2. This sits nicely between earlier parabolic results. They state the problem and the Holder regularity of a(x,t) clearly, and the results follow from the divergence structure under those gap conditions. The dependence on s and n in the second bound is the explicit new element. A minor soft spot is that the abstract gives no hint of the proof strategy, so I can't tell yet whether the interpolation requires fresh ideas or just careful bookkeeping with known tools like the parabolic Gehring lemma. From the stress test there is no obvious inconsistency or hidden assumption. This paper is for people deep in regularity theory for parabolic PDEs with nonstandard growth. A reader who follows work on double-phase problems will see the value in having these interpolated bounds available. I recommend sending it to peer review. The claims are focused and the setting is standard enough that referees can evaluate the details efficiently.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes gradient higher integrability results for weak solutions to the degenerate parabolic double-phase equation u_t - div(|Du|^{p-2}Du + a(x,t)|Du|^{q-2}Du) = 0 in Ω_T, where a ∈ C^{α, α/2}(Ω_T). For bounded solutions the result holds under the gap condition q ≤ p + α; for solutions satisfying u ∈ C(0,T; L^s(Ω)) with s ≥ 2 the gap condition is relaxed to q ≤ p + sα/(n+s). These bounds are presented as an interpolation between existing parabolic double-phase results.

Significance. If the proofs are correct, the work supplies a useful interpolation of gap conditions that refines the range of admissible (p,q,α) for which gradient higher integrability is known in the parabolic double-phase setting. The distinction between the bounded-solution case and the case with additional L^s integrability is a natural and potentially applicable refinement. The structural assumptions on a and the divergence-form operator are standard and clearly stated.

minor comments (2)

The abstract states the gap conditions cleanly, but the introduction would benefit from an explicit listing of the precise definition of weak solution (including the integrability class for Du) and the admissible ranges for p, q, α, n before the main theorems are stated.
Notation for the parabolic cylinder Ω_T and the Hölder space C^{α, α/2}(Ω_T) should be recalled or referenced at the beginning of the technical sections to aid readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. The recommendation for minor revision is noted, and we appreciate the recognition that our interpolation of gap conditions refines the admissible range for gradient higher integrability in the parabolic double-phase setting. Since no specific major comments were raised in the report, we address the overall summary below.

read point-by-point responses

Referee: The manuscript establishes gradient higher integrability results for weak solutions to the degenerate parabolic double-phase equation u_t - div(|Du|^{p-2}Du + a(x,t)|Du|^{q-2}Du) = 0 in Ω_T, where a ∈ C^{α, α/2}(Ω_T). For bounded solutions the result holds under the gap condition q ≤ p + α; for solutions satisfying u ∈ C(0,T; L^s(Ω)) with s ≥ 2 the gap condition is relaxed to q ≤ p + sα/(n+s). These bounds are presented as an interpolation between existing parabolic double-phase results.

Authors: We are grateful for the referee's concise and accurate summary of our main results. The distinction between the bounded-solution case (q ≤ p + α) and the case with additional temporal continuity in L^s (yielding the interpolated gap q ≤ p + sα/(n+s)) is indeed the key refinement we introduce, and we believe this interpolation is a natural and useful contribution to the literature on parabolic double-phase problems. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a proof of gradient higher integrability for weak solutions of the stated degenerate parabolic double-phase equation, relying on the given structural assumptions on the coefficient a and standard PDE techniques such as difference quotients, Gehring-type lemmas, and interpolation between integrability exponents. The gap conditions q ≤ p + α and q ≤ p + sα/(n+s) emerge from the estimates rather than being presupposed or fitted to the target result. No load-bearing step reduces by construction to a self-citation chain, a renamed input, or a parameter fitted to the output quantity itself. The derivation is self-contained against external benchmarks in the parabolic regularity literature.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions of parabolic PDE theory and Hölder regularity of coefficients; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Standard structural assumptions on p, q and the double-phase operator for weak solutions to exist and satisfy the stated equation.
Implicit in the setup of the degenerate parabolic double-phase problem.
domain assumption Hölder continuity of a(x,t) in space-time with exponent α.
Stated directly in the abstract as a(x,t) ∈ C^{α, α/2}(Ω_T).

pith-pipeline@v0.9.0 · 5443 in / 1255 out tokens · 27914 ms · 2026-05-17T20:45:34.774304+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

?

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

We establish gradient higher integrability results for weak solutions to degenerate parabolic equations of double phase type ut − div(|Du|^{p−2}Du + a(x,t)|Du|^{q−2}Du) = 0 … under the gap condition q ≤ p + α.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

These results provide an interpolation between the gap bounds in the parabolic double phase setting.

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

Works this paper leans on

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[1] [1]

Arora and S

R. Arora and S. Shmarev. Double-phase parabolic equations with variable growth and nonlinear sources.Adv. Nonlinear Anal., 12(1):304–335, 2023

work page 2023

[2] [2]

Baasandorj, S.-S

S. Baasandorj, S.-S. Byun, and J. Oh. Calder´ on-Zygmund estimates for gen- eralized double phase problems.J. Funct. Anal., 279(7):108670, 57, 2020

work page 2020

[3] [3]

Bahrouni, V

A. Bahrouni, V. t. D. R˘ adulescu, and D. s. D. Repovˇ s. Double phase tran- sonic flow problems with variable growth: nonlinear patterns and stationary waves.Nonlinearity, 32(7):2481–2495, 2019

work page 2019

[4] [4]

Baroni, M

P. Baroni, M. Colombo, and G. Mingione. Harnack inequalities for double phase functionals.Nonlinear Anal., 121:206–222, 2015

work page 2015

[5] [5]

Baroni, M

P. Baroni, M. Colombo, and G. Mingione. Regularity for general functionals with double phase.Calc. Var. Partial Differential Equations, 57(2):Paper No. 62, 48, 2018

work page 2018

[6] [6]

Benci, P

V. Benci, P. D’Avenia, D. Fortunato, and L. Pisani. Solitons in several space dimensions: Derrick’s problem and infinitely many solutions.Arch. Ration. Mech. Anal., 154(4):297–324, 2000

work page 2000

[7] [7]

K. O. Buryachenko and I. I. Skrypnik. Local continuity and Harnack’s in- equality for double-phase parabolic equations.Potential Anal., 56(1):137–164, 2022

work page 2022

[8] [8]

Byun and H.-S

S.-S. Byun and H.-S. Lee. Calder´ on-Zygmund estimates for elliptic double phase problems with variable exponents.J. Math. Anal. Appl., 501(1):Paper No. 124015, 31, 2021

work page 2021

[9] [9]

Byun and H.-S

S.-S. Byun and H.-S. Lee. Gradient estimates ofω-minimizers to double phase variational problems with variable exponents.Q. J. Math., 72(4):1191–1221, 2021. BOUNDED SOLUTIONS AND INTERPOLATIVE GAP BOUNDS 31

work page 2021

[10] [10]

Byun and J

S.-S. Byun and J. Oh. Global gradient estimates for non-uniformly elliptic equations.Calc. Var. Partial Differential Equations, 56(2):Paper No. 46, 36, 2017

work page 2017

[11] [11]

Byun and J

S.-S. Byun and J. Oh. Regularity results for generalized double phase func- tionals.Anal. PDE, 13(5):1269–1300, 2020

work page 2020

[12] [12]

Charkaoui, A

A. Charkaoui, A. Ben-Loghfyry, and S. Zeng. Nonlinear parabolic double phase variable exponent systems with applications in image noise removal.Appl. Math. Model., 132:495–530, 2024

work page 2024

[13] [13]

Y. Chen, S. Levine, and M. Rao. Variable exponent, linear growth functionals in image restoration.SIAM J. Appl. Math., 66(4):1383–1406, 2006

work page 2006

[14] [14]

Cherfils and Y

L. Cherfils and Y. Il’yasov. On the stationary solutions of generalized reaction diffusion equations withp&q-Laplacian.Commun. Pure Appl. Anal, 4(1):9–22, 2005

work page 2005

[15] [15]

Chlebicka, P

I. Chlebicka, P. Gwiazda, and A. Zatorska-Goldstein. Parabolic equation in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev’s phenomenon.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 36(5):1431–1465, 2019

work page 2019

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Colombo and G

M. Colombo and G. Mingione. Bounded minimisers of double phase variational integrals.Arch. Ration. Mech. Anal., 218(1):219–273, 2015

work page 2015

[17] [17]

Colombo and G

M. Colombo and G. Mingione. Regularity for double phase variational prob- lems.Arch. Ration. Mech. Anal., 215(2):443–496, 2015

work page 2015

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Colombo and G

M. Colombo and G. Mingione. Calder´ on-Zygmund estimates and non- uniformly elliptic operators.J. Funct. Anal., 270(4):1416–1478, 2016

work page 2016

[19] [19]

De Filippis and G

C. De Filippis and G. Mingione. A borderline case of Calder´ on-Zygmund estimates for nonuniformly elliptic problems.Algebra i Analiz, 31(3):82–115, 2019

work page 2019

[20] [20]

DiBenedetto.C 1+α local regularity of weak solutions of degenerate elliptic equations.Nonlinear Anal., 7(8):827–850, 1983

E. DiBenedetto.C 1+α local regularity of weak solutions of degenerate elliptic equations.Nonlinear Anal., 7(8):827–850, 1983

work page 1983

[21] [21]

Esposito, F

L. Esposito, F. Leonetti, and G. Mingione. Sharp regularity for functionals with (p, q) growth.J. Differential Equations, 204(1):5–55, 2004

work page 2004

[22] [22]

Giusti.Direct methods in the calculus of variations

E. Giusti.Direct methods in the calculus of variations. World Scientific Pub- lishing Co., Inc., River Edge, NJ, 2003

work page 2003

[23] [23]

Harjulehto and P

P. Harjulehto and P. H¨ ast¨ o. Double phase image restoration.J. Math. Anal. Appl., 501(1):Paper No. 123832, 12, 2021

work page 2021

[24] [24]

Harjulehto, P

P. Harjulehto, P. H¨ ast¨ o, V. Latvala, and O. Toivanen. Critical variable expo- nent functionals in image restoration.Appl. Math. Lett., 26(1):56–60, 2013

work page 2013

[25] [25]

H¨ ast¨ o and J

P. H¨ ast¨ o and J. Ok. Higher integrability for parabolic systems with Orlicz growth.J. Differential Equations, 300:925–948, 2021

work page 2021

[26] [26]

H¨ ast¨ o and J

P. H¨ ast¨ o and J. Ok. Maximal regularity for local minimizers of non-autonomous functionals.J. Eur. Math. Soc. (JEMS), 24(4):1285–1334, 2022

work page 2022

[27] [27]

H¨ ast¨ o and J

P. H¨ ast¨ o and J. Ok. Regularity theory for non-autonomous partial differ- ential equations without Uhlenbeck structure.Arch. Ration. Mech. Anal., 245(3):1401–1436, 2022

work page 2022

[28] [28]

Kbiri Alaoui, T

M. Kbiri Alaoui, T. Nabil, and M. Altanji. On some new non-linear diffusion models for the image filtering.Appl. Anal., 93(2):269–280, 2014

work page 2014

[29] [29]

Kim and J

B. Kim and J. Oh. Higher integrability for weak solutions to parabolic multi- phase equations.J. Differential Equations, 409:223–298, 2024. 32 BOGI KIM AND JEHAN OH

work page 2024

[30] [30]

B. Kim, J. Oh, and A. Sen. Parabolic Lipschitz truncation for multi-phase problems: the degenerate case.Adv. Calc. Var., 18(3):979–1010, 2025

work page 2025

[31] [31]

W. Kim. Calder´ on-Zygmund type estimate for the singular parabolic double- phase system.J. Math. Anal. Appl., 551(1):Paper No. 129593, 33, 2025

work page 2025

[32] [32]

W. Kim, J. Kinnunen, and K. Moring. Gradient higher integrability for degen- erate parabolic double-phase systems.Arch. Ration. Mech. Anal., 247(5):Paper No. 79, 46, 2023

work page 2023

[33] [33]

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