Logarithmic continuity for the Nonlocal degenerate two-phase Stefan problem

Harsh Prasad; Ho-Sik Lee; Kyeongbae Kim

REVIEW 1 major objections 5 minor 17 references

Logarithmic continuity for the Nonlocal degenerate two-phase Stefan problem

T0 review · 1 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Weak solutions of the nonlocal two-phase Stefan problem have a logarithmic modulus of continuity.

desk verdict The oscillation estimates are a solid advance, but Theorem 1.8's existence claim for merely continuous boundary data is not proved, and the paper needs a revision before it can be accepted. read the letter →

arxiv 2504.17383 v1 pith:4QYGGRAJ submitted 2025-04-24 math.AP

Kyeongbae Kim , Ho-Sik Lee , Harsh Prasad This is my paper · ORCID

classification math.AP MSC 35R0935A0135D3080A2235K65

keywords nonlocaltwo-phaseStefanproblemfractionalp-LaplacianlogarithmicmodulusofcontinuityintrinsicscalingoscillationestimatesboundaryweaksolutionsDeGiorgiiteration

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies the nonlocal two-phase Stefan equation, a model of ice–water phase transitions with fractional diffusion, and proves that weak solutions are continuous with an explicit logarithmic modulus, not merely qualitatively regular. For the regularized equation, the central theorem is an oscillation decay estimate in intrinsically scaled cylinders: $\operatorname{osc}_{Q_r} u_\varepsilon \le c\,\omega_0 (1+\ln(\rho_0/r))^{-\varsigma/2}+4\varepsilon$, uniformly in the regularization parameter $\varepsilon$. Passing to the limit in $\varepsilon$ produces a weak solution of the original singular Stefan problem that is continuous on $\Omega\times[0,T]$ with the quantitative bound $\sup_{z_1,z_2} |u(z_1)-u(z_2)| \le c\,[1+|\ln(1/(|x_1-x_2|+|t_1-t_2|^{1/(sp)}))|]^{-\varsigma/2}$. The result matters because it converts a qualitative question of well-posedness for anomalous phase transitions into a uniform, quantitative regularity statement that survives the singular limit.

What carries the argument

The central object is the sequence of intrinsic cylinders $Q_i=B_{\rho_i}(x_0)\times(t_0-\rho_i^{sp}(\omega_i/4)^{2-p},t_0]$, whose time scale is set by the current oscillation $\omega_i$, together with the iteration functions $f_1(\omega)=\omega^{M_1}/(N_1\omega_0^{M_1})$ and $f_2(\omega)=1-\omega^{M_2}/(N_2\omega_0^{M_2})$ that shrink radius and oscillation. The argument is carried by De Giorgi-type lemmas: under a small-tail condition, a measure-theoretic gain on a cylinder produces a pointwise gain on the next, smaller cylinder; a two-alternative argument yields the decay $\operatorname{osc}_{Q_i}u\le \omega_0(1+i)^{-\varsigma}+4\varepsilon$. The regularization $\beta_\varepsilon$ is a mollification of the maximal monotone graph $\beta$, and the estimates are deliberately independent of $\varepsilon$ so the singular limit can be taken.

What would settle it

Check directly whether the hypothesis (1.14) of Theorem 1.8 implies the bound (6.1) used in Lemma 6.1. A radial boundary datum $g(x)=(1+|\ln(1/|x|)|)^{-\delta/2}$ near a boundary point is continuous and time-independent, so it satisfies (1.14) locally, but it does not satisfy (6.1) with exponent $\delta$; if the proof cannot be modified to cover such a datum, the stated existence for the full class in (1.14) is not established.

Watch

Extended reading notes

Core claim

The paper's central claim is that, for $p>2$, $s\in(0,1)$, $n\ge 2$, and any symmetric measurable kernel $k$ with $\Lambda^{-1}\le k\le \Lambda$, solutions of the regularized two-phase Stefan equation satisfy the same kind of logarithmic oscillation decay that is known for the local Stefan problem, but now in a geometry adapted to the nonlocal operator. The time length of the cylinders is not fixed; it scales like $\rho^{sp}(\omega/4)^{2-p}$, where $\omega$ is the size of the solution on the cylinder. The same estimate holds at the lateral boundary, provided the complement of $\Omega$ satisfies a measure density condition, and at the initial boundary. Because all constants are independent of $\varepsilon$, a subsequence of regularized solutions converges to a weak solution of the original problem, and the estimate survives as the explicit logarithmic modulus in (1.15).

Load-bearing premise

The existence proof relies on the boundary datum satisfying a logarithmic modulus bound $\omega_g(r) \le c_g(1+|\ln(1/r)|)^{-\delta}$; the theorem as stated only assumes continuity and $\partial_t g\in L^2$, and no approximation step is given to close that gap.

Editorial extensions

If this is right

Interior and boundary continuity: any weak solution produced by the approximation is continuous on $\Omega\times[0,T]$ with modulus $(1+|\ln(1/r)|)^{-\varsigma/2}$, where the exponent is quantitative and depends only on the structural data.
The uniform-in-$\varepsilon$ estimates make the regularization scheme stable: no oscillation information is lost when $\varepsilon\to 0$, so the constructed solution is a genuine weak solution rather than a formal limit.
Boundary data with a logarithmic modulus are propagated: if $g$ has modulus $(1+|\ln(1/r)|)^{-\delta}$ with $\delta\in(\varsigma,1)$, the solution has the modulus in (1.15) with exponent $\varsigma/2$, with constants depending on the data and domain.
The result applies to the full parameter range $s\in(0,1)$, $p>2$, $n\ge 2$, and to all kernels comparable to the fractional $p$-Laplacian, so it covers a broad class of nonlocal phase-change models.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same tail-controlled De Giorgi iteration should imply Harnack-type estimates or time-insensitive Harnack inequalities for this nonlocal Stefan problem, since the intrinsic cylinders already encode the correct time scaling; the paper does not pursue this.
The exponent $\varsigma$ is produced by the iteration but never tracked; isolating its dependence on $n,s,p,\Lambda,\alpha_0$ would permit a sharpness comparison with the local $p$-Laplace modulus and with the improved moduli known for $p=n$.
The endpoint $\delta=\varsigma$ in the boundary datum is left open; the theorem requires $\delta>\varsigma$, and a natural test is whether the logarithmic modulus persists at the exact threshold.
A plausible extension is to nonlocal phase transitions with $p(x)$-dependent growth or to doubly nonlinear nonlocal operators, where intrinsic cylinders would need variable-exponent scaling; this is not addressed in the paper.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The interior and boundary oscillation estimates are a genuine advance. This is the first quantitative logarithmic modulus for the nonlocal two-phase Stefan problem with p>2, and the adaptation of intrinsic scaling and De Giorgi iteration from the local p-Laplace Stefan theory and fractional p-Laplacian parabolic equations is technically demanding and, as far as I can tell, coherent. Theorems 1.2, 1.4, and 1.5 give explicit oscillation decay in intrinsic cylinders, and the uniform estimates for the regularized solutions are the right vehicle for passing to a limit. I checked the main oscillation iteration in Section 3; no internal contradiction jumps out.

The soft spot is the advertised existence result. Theorem 1.8 states existence of a continuous weak solution for g merely continuous with ∂_t g ∈ L^2, but Lemma 6.1, which supplies the equicontinuity used in the Ascoli–Arzelá step, requires the boundary datum to satisfy the quantitative log-Hölder bound (6.1). The data class (1.14) does not imply (6.1), and no approximation argument is given. A naive approximation cannot produce uniform estimates because the constant in (6.2) depends on c_g and degrades as the modulus worsens. So the first sentence of Theorem 1.8 is unsupported. The fix is straightforward: state existence only under the log-Hölder assumption (6.1), which would still be a strong result. The secondary issue is minor: (1.15) is quantified 'for any r>0', but for r>1 the right-hand side decays while ω_g is nondecreasing; this should be restricted to the diameter range.

Several technical lemmas are imported from previous papers, some by the same authors. That is not a defect in itself, but it makes independent verification harder. Nothing I saw suggests those lemmas are wrong.

This paper is for specialists in nonlocal parabolic regularity. The quantitative estimates are likely to be used and cited. The existence gap is real but fixable. It deserves a serious referee; I would send it out with a request to revise Theorem 1.8 and clarify the range of r in (1.15).

Referee Report

1 major / 5 minor

Summary. The paper studies the nonlocal degenerate two-phase Stefan problem ∂t(u+β(u))+Lu ∋ 0, where L is a fractional p-Laplace type operator with p>2 and a measurable kernel. For the regularized problem with smooth monotone βε, the authors prove quantitative oscillation decay in intrinsic cylinders: interior estimates (Theorem 1.2), lateral boundary estimates (Theorem 1.4), and initial boundary estimates (Theorem 1.5). These are combined in Lemma 6.1 into a global logarithmic modulus of continuity for the regularized solutions, and an Ascoli-Arzelá compactness argument is used to construct a continuous weak solution of the original problem and to claim a quantitative logarithmic modulus (Theorem 1.8). The proof strategy follows the intrinsic-scaling method of BKU14 and Lia22, adapted to the nonlocal setting with tail controls and De Giorgi-type lemmas.

Significance. If the main estimates are correct, the paper would provide the first quantitative logarithmic modulus of continuity for this nonlocal degenerate two-phase Stefan problem, extending the local results of BKU14 and Lia22 to the fractional p>2 setting and complementing the existence result of ACM22. The technical core is substantial: the authors construct explicit sequences of intrinsic cylinders, prove De Giorgi-type lemmas with tail controls, and track the dependence of constants on the data. These are genuine strengths. However, the advertised existence theorem for merely continuous boundary data is not supported by the proof, because the uniform modulus used in the compactness argument requires a log-Hölder boundary datum. This is a load-bearing gap, although it is localized to Theorem 1.8 and could be repaired by strengthening the hypotheses or by supplying a genuinely new approximation argument.

major comments (1)

[Section 6, Lemma 6.1 and Theorem 1.8] The existence assertion for the data class (1.14) is not established by the proof. Lemma 6.1, which supplies the uniform modulus (6.2) used in the Ascoli-Arzelá step of Theorem 1.8, is conditional on hypothesis (6.1): the boundary datum must satisfy ω_g(ρ) ≤ c_g(1+|ln(1/ρ)|)^{-δ} with δ ∈ (ς,1). The hypotheses in (1.14) only require g ∈ C([0,T]×Ω') ∩ L^p(0,T;W^{s,p}(Ω')) ∩ L^∞(0,T;L^∞(R^n)) with ∂_t g ∈ L^2(Ω_T), which does not imply (6.1). No approximation of a merely continuous g by log-Hölder data is supplied, and a naive approximation cannot yield uniform equicontinuity because the constant in (6.2) depends on c_g through the choice of N_0 in (4.28). Thus the first assertion of Theorem 1.8, existence of a weak solution u ∈ C(Ω×[0,T]) for the advertised class, is unproved.

minor comments (5)

[Abstract] The phrase 'We study the the problem' contains a duplicated article and should read 'We study the problem'.
[Theorem 1.8, Eq. (1.15)] The modulus estimate is stated for arbitrary r>0, but for r>1 the right-hand side decays to zero as r→∞, whereas the left-hand side over a pair with that separation, when such a pair exists, is controlled below by the oscillation of g if g is nonconstant. Since Ω×[0,T] is compact, the constant c could absorb the finite range of r, but the quantification should be restricted explicitly (for example to r<1) to avoid an apparent inconsistency.
[Lemma 2.2] The symbol ε is used both for the regularization parameter in (1.5) and for the auxiliary small constant in the sequence a_n; this collision is confusing and the auxiliary constant should be renamed.
[Remark 1.6] The remark asserts that the normalization assumptions (1.9) and (1.11) can be removed by following Sections 4 and 5, but no proof of the unnormalized variants is given. If these variants are needed for the global estimate in Lemma 6.1, the exact statements and their proofs should be included.
[Sections 3–5] Several technical lemmas are quoted from BKU14, Lia24c, DZZ21, and BLS21 without restating their hypotheses; for a self-contained journal version, the precise statements used should be recalled or referenced with the exact assumptions needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: estimates are independently derived; self-citations are not load-bearing.

full rationale

The paper's central claims are oscillation estimates for solutions of a regularized nonlocal two-phase Stefan problem (Theorems 1.2, 1.4, 1.5) and an existence-with-modulus theorem (Theorem 1.8). These are derived by constructing intrinsic cylinders, proving De Giorgi-type measure decay lemmas (Lemmas 3.4, 3.6, 4.2, 4.3), and iterating oscillation reductions. The induction steps are fully written out; the target estimate is not assumed at any point. Self-citations (APT24, DKLN25, BK24, Pra24) appear only as references for standard energy estimates, tail estimates, or the technique of intrinsic cylinders; the load-bearing De Giorgi lemmas are proved in the paper or cited from non-overlapping authors (Lia24c, DZZ21, BKLU18). No parameter is fitted to the conclusion, and no uniqueness theorem is imported from the authors' prior work. The proof is therefore self-contained in the sense required for circularity analysis. A separate correctness caveat: Theorem 1.8's first assertion (existence for g merely continuous with ∂_t g ∈ L^2) is not established by the proof, because the uniform equicontinuity estimate (6.2) used in the Ascoli–Arzelá argument is conditional on the log-Hölder hypothesis (6.1) of Lemma 6.1, and the data class (1.14) does not imply (6.1); no approximation argument is supplied. This is a proof gap, not a circular step; likewise the r>0 quantification in (1.15) is meaningful only for r<1 for nonconstant g. These issues do not affect the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No empirical parameters are fitted; the central claim depends on structural PDE hypotheses (kernel bounds, measure density, log-Holder boundary data) and on a chain of standard lemmas from prior literature. The exponents xi and delta are universal constants determined by n,s,p,Lambda,alpha_0 and the data modulus, not adjustable fit parameters. No new physical entities are postulated.

assumptions (4)

domain assumption Measure density condition (1.7): |B_r(x0) \setminus Omega| >= alpha_0 |B_r| for all x0 in partial Omega and 0<r<=1.
Invoked in Lemmas 4.2 and 4.3 for the lateral boundary De Giorgi argument and in Theorem 1.8 to control oscillations near partial Omega. It is standard but restrictive.
domain assumption Boundary data regularity (1.14) and log-modulus condition (6.1): g in C([0,T] x Omega') cap L^p(0,T;W^{s,p}(Omega')) cap L^infinity(0,T;L^infinity(R^n)), partial_t g in L^2(Omega_T), and omega_g(r) <= c_g(1+|ln(1/r)|)^{-delta} with delta in (xi,1).
Needed for the uniform equicontinuity input in Lemma 6.1 and hence for the quantitative modulus (1.15). The theorem's broader existence statement for merely continuous g is not supported by the proof as written.
domain assumption Kernel symmetry and bounds: k(x,y,t)=k(y,x,t) and Lambda^{-1} <= k <= Lambda.
Used throughout all energy estimates and De Giorgi lemmas to keep the nonlocal operator comparable to the fractional p-Laplacian. This is a standard structural assumption.
standard math Prior regularity and existence lemmas used as black boxes: BKU14 Lemma 2.1, Lia24c Lemmas 3.1-3.4, DZZ21 Lemma 2.3, BLS21 Theorem A.3 and Proposition A.4, and Coz17 Lemma 4.7.
The proof imports Caccioppoli estimates, De Giorgi-type propagation lemmas, fractional Sobolev embeddings, solvability of the regularized problem, and a fractional Poincare inequality. If any of these are inapplicable under the stated hypotheses, the estimates would collapse.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Logarithmic continuity for the Nonlocal degenerate two-phase Stefan problem." pith.science (2026). https://pith.science/paper/4QYGGRAJ

@misc{pith2026250417383,
  author       = {Pith},
  title        = {Pith review of: Logarithmic continuity for the Nonlocal degenerate two-phase Stefan problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4QYGGRAJ}},
  note         = {Machine review of arXiv:2504.17383}
}

read the original abstract

We establish certain oscillation estimates for weak solutions to nonlinear, anomalous phase transitions modeled on the nonlocal two-phase Stefan problem. The problem is singular in time, is scaling deficient and influenced by far-off effects. We study the the problem in a geometry adapted to the solution and obtain oscillation estimates in intrinsically scaled cylinders. Furthermore, via certain uniform estimates, we construct a continuous weak solution to the corresponding initial boundary value problem with a quantitative modulus of continuity.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 10 canonical work pages

[1]

T he two-phase Stefan problem with anoma- lous diﬀusion

[ACM22] I. Athanasopoulos, L. Caﬀarelli, and E. Milakis. “T he two-phase Stefan problem with anoma- lous diﬀusion”. In: Adv. Math. 406 (2022), Paper No. 108527,

work page 2022
[8]

Local boundedness of solutions to parabolic equations associated with fractional $p$-Laplacian type operators

arXiv: 2412.03770 [math.AP] . [Lia22] N. Liao. “On the logarithmic type boundary modulus o f continuity for the Stefan problem: to the memory of Emmanuele DiBenedetto”. In: Adv. Math. 408 (2022), Paper No. 108613,

work page Pith review arXiv 2022
[12]

Po tential theory for nonlocal drift- diﬀusion equations

arXiv: 2409.20097 [math.AP] . [NNSW24] Q.-H. Nguyen, S. Nowak, Y. Sire, and M. Weidner. “Po tential theory for nonlocal drift- diﬀusion equations”. In: Arch. Ration. Mech. Anal. 248.6 (2024), Paper No. 126,

arXiv 2024
[14]

Local boundedness of variat ional solutions to nonlocal dou- ble phase parabolic equations

doi: 10.1007/s00526-024-02670-3 . [PT23] H. Prasad and V. Tewary. “Local boundedness of variat ional solutions to nonlocal dou- ble phase parabolic equations”. In: J. Diﬀerential Equations 351 (2023), pp. 243–276. doi: 10.1016/j.jde.2022.12.029. [RD23] S. Rogosin and M. Dubatovskaya. “Fractional Stefan p roblem: a survey of the recent results”. In: Lobach...

work page doi:10.1007/s00526-024-02670-3 2023
[15]

Harnack’s inequality for parabo lic nonlocal equations

[Str19a] M. Str¨ omqvist. “Harnack’s inequality for parabo lic nonlocal equations”. In: Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire36.6 (2019), pp. 1709–1745. doi: 10.1016/j.anihpc.2019.03.003. [Str19b] M. Str¨ omqvist. “Local boundedness of solutions t o non-local parabolic equations modeled on the fractional p-Laplacian”. In: J. Diﬀerential Equation...

work page doi:10.1016/j.anihpc.2019.03.003 2019
[19]

Local H¨ old er regularity for nonlocal parabolic p-Laplace equations

doi: 10.1016/j.aim.2022.108527. [APT24] K. Adimurthi, H. Prasad, and V. Tewary. “Local H¨ old er regularity for nonlocal parabolic p-Laplace equations”. In: arXiv:2205.09695 (2024). To appear in Ann. Sc. Norm. Super. Pisa CI. Sci. (5). arXiv: 2205.09695 [math.AP] . [BGK23] A. Banerjee, P. Garain, and J. Kinnunen. “Lower semi continuity and pointwise behav...

arXiv 2024
[23]

A H¨ older estimate with an opti mal tail for nonlocal parabolic p- Laplace equations

doi: 10.1142/S0219199722500328. [BK24] S.-S. Byun and K. Kim. “A H¨ older estimate with an opti mal tail for nonlocal parabolic p- Laplace equations”. In: Ann. Mat. Pura Appl. (4) 203.1 (2024), pp. 109–147. doi: 10.1007/s10231-023-01355-6 . [BKLU18] P. Baroni, T. Kuusi, C. Lindfors, and J. M. Urbano. “ Existence and boundary regularity for degenerate phas...

work page doi:10.1142/s0219199722500328 2024
[29]

The parabolic Harnack in equality for nonlocal equations

doi: 10.1007/s00526-016-0999-2 . [KW23] M. Kassmann and M. Weidner. “The parabolic Harnack in equality for nonlocal equations”. In: arXiv:2303.05975 (2023). arXiv: 2303.05975. [KWZ24] T. Kumagai, J. Wang, and M.-g. Zhang. Local boundedness of solutions to parabolic equations associated with fractional p-Laplacian type operators

arXiv 2023

Show all 17 references

[30]

[L W24] N

doi: 10.1112/jlms.12985. [L W24] N. Liao and M. Weidner. Time-insensitive nonlocal parabolic Harnack estimates

doi
[34]

On the modulus of continuity of solutions to nonlocal parabolic equations

doi: 10.1007/s00526-023-02627-y . [Lia24d] N. Liao. “On the modulus of continuity of solutions to nonlocal parabolic equations”. In: J. Lond. Math. Soc. (2) 110.3 (2024), Paper No. e12985,

2024 doi
[42]

A method of solution of the general S tefan problem

doi: 10.1007/s00205-024-02073-w . [Ole60] O. A. Ole ˘ ınik. “A method of solution of the general S tefan problem”. In: Soviet Math. Dokl. 1 (1960), pp. 1350–1354. [Pra24] H. Prasad. “On the weak Harnack estimate for nonloca l equations”. In: Calc. Var. Partial Diﬀerential Equa...

1960 doi
[45]

Local regularity for p arabolic nonlocal operators

doi: 10.1007/s00526-020-01870-x . [FK13] M. Felsinger and M. Kassmann. “Local regularity for p arabolic nonlocal operators”. In: Comm. Partial Diﬀerential Equations 38.9 (2013), pp. 1539–1573. doi: 10.1080/03605302.2013.808211. [GLT25] P. Garain, E. Lindgren, and A. Tavakoli. ...

2013
[53]

An improved modulus of continuity for the two-phase Stefan problem

doi: 10.1016/j.aim.2022.108613. [Lia24a] N. Liao. “An improved modulus of continuity for the two-phase Stefan problem”. In: Trans. Amer. Math. Soc. 377.9 (2024), pp. 6023–6041. doi: 10.1090/tran/9093. [Lia24b] N. Liao. “H¨ older estimates for the Stefan problem ”. In: SIAM J. ...

2024
[76]

Continuous solutions for a degenerat e free boundary problem

doi: 10.1007/s00028-024-00949-8 . [Urb00] J. M. Urbano. “Continuous solutions for a degenerat e free boundary problem”. In: Ann. Mat. Pura Appl. (4) 178 (2000), pp. 195–224. doi: 10.1007/BF02505895. [Urb08] J. M. Urbano. The method of intrinsic scaling . Vol

2000 doi
[514]

The obs tacle problem for nonlinear integro- diﬀerential operators

REFERENCES 35 [KKP16] J. Korvenp¨ a¨ a, T. Kuusi, and G. Palatucci. “The obs tacle problem for nonlinear integro- diﬀerential operators”. In: Calc. Var. Partial Diﬀerential Equations 55.3 (2016), Art. 63,

2016
[1930]

A free boundary problem with convecti on for the p-Laplacian

Lecture Notes in Mathematics. A systematic approach to regularity for degenerate and singu lar PDEs. Springer-Verlag, Berlin, 2008, pp. x+150. doi: 10.1007/978-3-540-75932-4 . [Urb97] J. M. Urbano. “A free boundary problem with convecti on for the p-Laplacian”. In: Rend. Mat. ...

1997 doi
[2024]

On Stefan’s problem

arXiv: 2408.11555 [math.AP] . [Kam61] S. L. Kamenomostskaja. “On Stefan’s problem”. In: Mat. Sb. (N.S.) 53(95) (1961), pp. 489–

1961 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

T he two-phase Stefan problem with anoma- lous diﬀusion

[ACM22] I. Athanasopoulos, L. Caﬀarelli, and E. Milakis. “T he two-phase Stefan problem with anoma- lous diﬀusion”. In: Adv. Math. 406 (2022), Paper No. 108527,

work page 2022

[2] [8]

Local boundedness of solutions to parabolic equations associated with fractional $p$-Laplacian type operators

arXiv: 2412.03770 [math.AP] . [Lia22] N. Liao. “On the logarithmic type boundary modulus o f continuity for the Stefan problem: to the memory of Emmanuele DiBenedetto”. In: Adv. Math. 408 (2022), Paper No. 108613,

work page Pith review arXiv 2022

[3] [12]

Po tential theory for nonlocal drift- diﬀusion equations

arXiv: 2409.20097 [math.AP] . [NNSW24] Q.-H. Nguyen, S. Nowak, Y. Sire, and M. Weidner. “Po tential theory for nonlocal drift- diﬀusion equations”. In: Arch. Ration. Mech. Anal. 248.6 (2024), Paper No. 126,

arXiv 2024

[4] [14]

Local boundedness of variat ional solutions to nonlocal dou- ble phase parabolic equations

doi: 10.1007/s00526-024-02670-3 . [PT23] H. Prasad and V. Tewary. “Local boundedness of variat ional solutions to nonlocal dou- ble phase parabolic equations”. In: J. Diﬀerential Equations 351 (2023), pp. 243–276. doi: 10.1016/j.jde.2022.12.029. [RD23] S. Rogosin and M. Dubatovskaya. “Fractional Stefan p roblem: a survey of the recent results”. In: Lobach...

work page doi:10.1007/s00526-024-02670-3 2023

[5] [15]

Harnack’s inequality for parabo lic nonlocal equations

[Str19a] M. Str¨ omqvist. “Harnack’s inequality for parabo lic nonlocal equations”. In: Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire36.6 (2019), pp. 1709–1745. doi: 10.1016/j.anihpc.2019.03.003. [Str19b] M. Str¨ omqvist. “Local boundedness of solutions t o non-local parabolic equations modeled on the fractional p-Laplacian”. In: J. Diﬀerential Equation...

work page doi:10.1016/j.anihpc.2019.03.003 2019

[6] [19]

Local H¨ old er regularity for nonlocal parabolic p-Laplace equations

doi: 10.1016/j.aim.2022.108527. [APT24] K. Adimurthi, H. Prasad, and V. Tewary. “Local H¨ old er regularity for nonlocal parabolic p-Laplace equations”. In: arXiv:2205.09695 (2024). To appear in Ann. Sc. Norm. Super. Pisa CI. Sci. (5). arXiv: 2205.09695 [math.AP] . [BGK23] A. Banerjee, P. Garain, and J. Kinnunen. “Lower semi continuity and pointwise behav...

arXiv 2024

[7] [23]

A H¨ older estimate with an opti mal tail for nonlocal parabolic p- Laplace equations

doi: 10.1142/S0219199722500328. [BK24] S.-S. Byun and K. Kim. “A H¨ older estimate with an opti mal tail for nonlocal parabolic p- Laplace equations”. In: Ann. Mat. Pura Appl. (4) 203.1 (2024), pp. 109–147. doi: 10.1007/s10231-023-01355-6 . [BKLU18] P. Baroni, T. Kuusi, C. Lindfors, and J. M. Urbano. “ Existence and boundary regularity for degenerate phas...

work page doi:10.1142/s0219199722500328 2024

[8] [29]

The parabolic Harnack in equality for nonlocal equations

doi: 10.1007/s00526-016-0999-2 . [KW23] M. Kassmann and M. Weidner. “The parabolic Harnack in equality for nonlocal equations”. In: arXiv:2303.05975 (2023). arXiv: 2303.05975. [KWZ24] T. Kumagai, J. Wang, and M.-g. Zhang. Local boundedness of solutions to parabolic equations associated with fractional p-Laplacian type operators

arXiv 2023