{"id":"66cfde3a-3eb8-4ef9-9cac-509b2e99fae5","arxiv_id":"2504.17383","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Weak solutions of the nonlocal p>2 two-phase Stefan problem have a logarithmic modulus of continuity up to the boundary, and a continuous weak solution of the initial-boundary value problem exists.","lead":"The authors prove quantitative logarithmic continuity estimates for weak solutions of a nonlocal, degenerate two-phase Stefan problem with long-range diffusion. The result supplies a rigorous existence theorem for continuous weak solutions, supporting mathematical models of melting and freezing with anomalous, nonlocal heat transport.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.8's existence assertion for merely continuous boundary data is unproved: the only uniform estimate used, Lemma 6.1, requires log-Hölder data, and no approximation argument is given.","rationale":"I read the main oscillation lemmas as a coherent extension of De Giorgi and intrinsic-scaling techniques; I found no internal contradiction in Theorems 1.2, 1.4, or 1.5. The weakest point is the passage from uniform estimates for regularized solutions to existence for the original problem in Theorem 1.8. The proof's only compactness input is (6.2), and (6.2) is gated by the log-Hölder hypothesis (6.1). The theorem advertises a broader data class, and no approximation argument is given; moreover, the dependence of the constants through N0 in (4.28) indicates that approximating continuous data by log-Hölder data would not produce a uniform estimate. This is exactly the reader's weakest assumption, and it warrants keeping the verdict CONDITIONAL. The additional quantification issue in (1.15) for r>1 is a statement-level defect that should be fixed by restricting to r<1, but it is secondary to the missing existence argument. Overall, the central quantitative estimates may well be correct, but the existence theorem as stated is not fully proven.","tokens_in":958,"tokens_out":3181,"duration_ms":142820,"concrete_test":"Re-run the proof of Theorem 1.8 with a boundary datum that is continuous, has ∂_t g∈L^2, but is not log-Hölder, for example g(x,t)=1/log(1/|x-x0|) near a boundary point x0. Verify whether Lemma 6.1 applies; it does not, because (6.1) fails. Then attempt the natural fix: approximate g by log-Hölder functions g_j and track constants. From (4.28), N0 must satisfy c_g(1+(i+1)ln N0)^{-δ} ≤ (2+i)^{-δ/2}; hence N0→∞ as c_j→∞, and the constant in (6.2) has no bound independent of j. This shows the Ascoli–Arzelá compactness step in Theorem 1.8 cannot be carried out for the stated class without a new argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.8's first assertion is not supported by the proof. The compactness step in the proof of Theorem 1.8 invokes the Ascoli–Arzelá argument together with Lemma 6.1's estimate (6.2). Lemma 6.1, however, is conditional on hypothesis (6.1): the boundary datum must satisfy ω_g(ρ) ≤ c_g(1+|ln(1/ρ)|)^{-δ} for some δ∈(ς,1). The theorem's data class (1.14) only asks for 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 a uniform estimate: the constant in (6.2) depends on c_g through the choice of N0 in (4.28), and N0 grows with c_g. Hence the uniform-in-ε equicontinuity needed to pass to a continuous limit u is missing for the advertised data class. A secondary statement issue is that (1.15) quantifies 'for any r>0', but for nonconstant g the right-hand side tends to 0 as r→∞ while ω_g is nondecreasing; the modulus claim only makes sense for r<1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":39585,"tokens_out":9098,"duration_ms":88650,"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":[{"comment":"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.","section":"Section 6, Lemma 6.1 and Theorem 1.8"}],"minor_comments":[{"comment":"The phrase 'We study the the problem' contains a duplicated article and should read 'We study the problem'.","section":"Abstract"},{"comment":"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.","section":"Theorem 1.8, Eq. (1.15)"},{"comment":"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.","section":"Lemma 2.2"},{"comment":"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.","section":"Remark 1.6"},{"comment":"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.","section":"Sections 3–5"}],"recommendation":"major_revision","confidential_remarks":"The technical oscillation machinery in Sections 3–5 appears coherent, and I did not find internal circularity in the main estimates. The serious problem is the mismatch between the data class advertised in Theorem 1.8 and the log-Hölder assumption actually required by Lemma 6.1. This is a load-bearing issue in the paper's headline existence result, and it should be fixed by reformulating Theorem 1.8 with the stronger hypothesis or by providing a valid approximation argument. I would also suggest asking the authors to state explicitly which results are being imported from overlapping-author papers, as this would help referees verify the proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nSeveral 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.\n\nThis 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).","headline":"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.","tokens_in":40170,"tokens_out":4294,"would_cite":true,"duration_ms":41885,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R09","35A01","35D30","80A22","35K65"],"pacs":[],"model":"deepseek-v4-flash","headline":"Weak solutions of the nonlocal two-phase Stefan problem have a logarithmic modulus of continuity.","keywords":["nonlocal two-phase Stefan problem","fractional p-Laplacian","logarithmic modulus of continuity","intrinsic scaling","oscillation estimates","boundary continuity","weak solutions","De Giorgi iteration"],"falsifier":"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.","tokens_in":39095,"feed_emoji":"❄️","tokens_out":10792,"duration_ms":95205,"temperature":0.7,"pith_summary":"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.","feed_headline":"Logarithmic continuity proved for nonlocal two-phase Stefan problem","feed_subtitle":"Uniform oscillation estimates on intrinsic cylinders make the singular phase-change limit continuous.","key_machinery":"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.","core_discovery":"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).","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the quantitative modulus-of-continuity template for the local two-phase Stefan problem that the interior oscillation proof adapts.","marker":"[BKU14]"},{"why":"provides the boundary logarithmic-modulus framework and the De Giorgi lemmas used for the lateral boundary estimates.","marker":"[Lia22]"},{"why":"gives the existence and boundary-regularity scheme for degenerate phase transitions whose approximation argument is followed in Theorem 1.8.","marker":"[BKLU18]"},{"why":"supplies the energy estimates and De Giorgi lemmas for the parabolic fractional p-Laplacian that are used when the singular term plays no role.","marker":"[Lia24c]"},{"why":"shows how to construct intrinsic cylinders for parabolic fractional p-Laplace equations, used to control the tail on each cylinder.","marker":"[APT24]"},{"why":"establishes continuity for the two-phase Stefan problem with anomalous diffusion and provides the baseline this paper extends to an explicit modulus.","marker":"[ACM22]"},{"why":"provides existence, uniqueness, and the maximum principle for the regularized nonlocal problem that the paper solves with the given boundary data.","marker":"[BLS21]"},{"why":"contains the iteration lemma that converts the algebraic recursion on oscillation sizes into decay.","marker":"[DiB93]"},{"why":"supplies the fractional Sobolev embedding and Poincaré inequality under the measure density condition, used for uniform energy bounds.","marker":"[Coz17]"}],"fun_headline_variants":["Log continuity proven for nonlocal two-phase Stefan","Nonlocal Stefan oscillations now logarithmically controlled","Intrinsic cylinders unlock logarithmic continuity in nonlocal Stefan","Quantitative moduli for nonlocal degenerate phase transitions","Singular nonlocal Stefan solved with logarithmic regularity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Log continuity proven for nonlocal two-phase Stefan","Nonlocal Stefan oscillations now logarithmically controlled","Intrinsic cylinders unlock logarithmic continuity in nonlocal Stefan","Quantitative moduli for nonlocal degenerate phase transitions","Singular nonlocal Stefan solved with logarithmic regularity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000339,"raw_usage":{"total_tokens":1793,"prompt_tokens":788,"completion_tokens":1005,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":404,"completion_tokens_details":{"reasoning_tokens":933}},"tokens_in":404,"tokens_out":1005,"duration_ms":9357,"temperature":1.0,"reasoning_tokens":933,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:42:46.776710+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}