{"id":"4ce52027-8d3d-4340-9b7c-b5e20f791f60","arxiv_id":"2507.15722","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Bounded weak solutions to parabolic p-Laplace systems with Hölder continuous coefficients have locally Hölder continuous spatial gradients for every p > 1, with quantitative estimates in terms of the solution's oscillation.","lead":"This paper proves that the slopes of solutions to a family of spreading equations remain smooth (Hölder continuous) over space and time, for every spreading exponent p > 1, a case that was previously open for systems below a critical exponent. The same result yields precise smoothness and decay estimates for fast diffusion equations, which model flows in porous media and other slow-spreading phenomena.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 5.1 is the load-bearing step: it imports [11, Thm 1.3] and asserts without proof that the radius restriction can be dropped for time-only-dependent coefficients; if wrong, the freezing argument and Theorem 1.1 collapse for the full range p>1.","rationale":"The reader's weakest_assumption identified exactly the same load-bearing concern: the unproved removal of the radius restriction in Proposition 5.1, which is imported from the authors' own [11]. My independent reading confirms this is the single most fragile point. The theorem's conclusion, including the quantitative bounds (1.3)-(1.4), is obtained by first proving a priori estimates under the assumption that the gradient is bounded and then passing to the limit. The a priori estimates in Sections 5.2 and 5.3 all rely on the Campanato-type estimate (5.9); in fact, (5.20) and (5.21) are direct consequences, and Proposition 5.6 converts them into the final Hölder estimate. If (5.9) fails for some p or for radii near the allowed upper bound, the interpolation argument in Section 5.2 (which explicitly uses R/R_o = 8^κ(r/R_o)^κ and lets R approach R_o) does not close, and Theorem 1.1 is not established. The concern is not that the result is false, but that a critical assertion in the proof is currently unsupported. The proof of Proposition 5.1 consists solely of a reference to a published paper plus a one-sentence claim about the radius restriction; that claim is precisely the kind of step that needs independent verification. I also note the secondary dependency on [8] for the doubly nonlinear application, but that is downstream and explicitly disclosed. The verdict CONDITIONAL is appropriate: the paper is credible and mostly self-contained, but it should be accepted only after Proposition 5.1's radius-removal claim is verified, either by a written proof or by pointing to the exact argument in [11] that guarantees it. I agree with the reader's assessment that this is the weakest assumption, and I recommend the same conditional status.","tokens_in":76667,"tokens_out":6536,"duration_ms":74163,"concrete_test":"Extract [11, Theorem 1.3] and its proof; locate the exact condition defining the radius bound ρ_o and determine whether it arises from x-Hölder continuity of the coefficients or from the intrinsic time geometry and the sup-estimate. Then independently re-derive Step 1 of [11] for coefficients b(t) independent of x on cylinders Q_r^(λ) with r up to R_o, checking whether the sup-bound |Dv| ≤ λ̃_μ holds without any smallness condition on R. If that is inconclusive, run a numerical check for N=2, p=1.5, b≡1, μ=0, λ=1: compute the L^p mean oscillation of ∇u on Q_r^(λ) for r/R = 0.25, 0.5, 0.75, 0.9, and verify that the ratio (r/R)^{-βp} mean-osc remains bounded with a fixed β>0 as r→R; failure of uniformity would disprove Proposition 5.1.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The entire Schauder machinery in Section 5 rests on Proposition 5.1, which supplies the Campanato-type estimate (5.9) for solutions of the frozen problem (5.7) on fixed-geometry intrinsic cylinders Q_r^(λ), uniformly for all r in (0,R] and all p>1. The proof is a rescaling v = A^{-1}w, followed by an invocation of [11, Theorem 1.3] and the sentence: 'Since the coefficients b(t) are independent of x, the upper bound ρ_o on the radius R can be avoided in the present situation.' No derivation is given that the radius restriction in [11] is exclusively an x-dependence artifact. In [11] the estimate is proved on cylinders with time scale (μ̃² + λ̃_μ²)^{(2-p)/2} τ² = λ^{2-p} τ², and the identification (5.10) is algebraically correct. However, the radius bound ρ_o in [11] could equally originate from the parabolic sup-estimate (Step 1), where the spatial radius must be controlled relative to the intrinsic time scale and the degeneracy of the p-Laplacian; x-independence of the coefficients does not remove that requirement. The critical use in Section 5.2 allows the intermediate radius R to approach R_o as r approaches R_o/8, so a bound on the absolute radius R matters, not only the ratio τ/R. If (5.9) fails or the constants degrade for radii near ρ_o, then inequalities (5.20), (5.21), and the gradient Hölder estimate in Proposition 5.6 and Theorem 1.1 are unproved, especially for p close to 1 where the intrinsic time scale λ^{2-p} is most sensitive. The paper itself flags related dependencies on the unreviewed preprint [8] for the application in Section 6, but Proposition 5.1 is internal to the main theorem and is asserted rather than demonstrated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript establishes local Hölder regularity of the spatial gradient for bounded weak solutions to the parabolic p-Laplace system ∂t u − div(a(x,t)(μ²+|Du|²)^{(p−2)/2}Du)=0 under the assumption that a is bounded, bounded away from zero, and Hölder continuous in the spatial variable, for every p>1 and μ∈[0,1]. The main theorem gives quantitative local L∞-gradient bounds and a local gradient Hölder estimate in terms of the oscillation of u, with constants depending only on N, p, C_o, C_1, α, and k. The proof combines self-contained energy estimates, Moser and De Giorgi iterations, comparison estimates for frozen-coefficient problems, and a Campanato-type freezing argument. As an application, the authors derive gradient estimates for a doubly nonlinear parabolic equation in a super-critical fast diffusion regime. The paper is transparent about the fact that in the sub-critical range 11, and it supplies explicit quantitative bounds rather than only qualitative regularity. The application to doubly nonlinear equations in Section 6 is nontrivial and gives concrete new estimates, including decay estimates at the extinction time. The paper is also unusually explicit about structural constants and about the limitations in the sub-critical range, and it contains long, detailed energy and comparison arguments that are valuable in themselves. However, the central claim currently rests on an imported radius-free Campanato estimate in Proposition 5.1 whose proof is only sketched by referring to the authors' earlier work [11]. The gap is localized, but it is load-bearing for the freezing argument and therefore for Theorem 1.1 and Section 6.","major_comments":[{"comment":"The proof of Proposition 5.1 invokes [11, Theorem 1.3] after the rescaling v=A^{-1}w and then asserts that 'since the coefficients b(t) are independent of x, the upper bound ρ_o on the radius R can be avoided in the present situation.' No derivation is given for this assertion. The radius restriction in [11] may be tied to the parabolic sup-estimate and to the intrinsic coupling sup(|Dw|²+μ²)^{1/2}≤Aλ, neither of which disappears when the coefficient depends only on time. This is load-bearing because (5.9) is the only input for the Campanato estimates (5.17), (5.20), and (5.21), which in turn feed Proposition 5.3, Proposition 5.6, and Theorem 1.1. The statement of Proposition 5.1 also contains an impossible hypothesis: 'if Q^{(λ)}_{2R}(z_o) ⋐ Q^{(λ)}_R(z_o)' cannot hold for R>0, and the condition 'R∈(0,1/2 R)' is circular. The authors should either reproduce the relevant steps of [11, Section 5.2] in the time-only case, or state and prove the radius-free version as a self-contained lemma, and correct the cylinder inclusion in the statement.","section":"§5.1, Proposition 5.1"},{"comment":"In the sub-quadratic case 11, either Corollary 3.9 must be reformulated to allow λ≥μ/A with constants depending on A, or the argument must first rescale the solution. This matters because (5.14) is used to derive the intrinsic coupling (5.16), which in turn is needed to apply Proposition 5.1 and obtain (5.17). This gap is technical and likely fixable, but it must be closed before the estimates in Section 5.2 are fully justified.","section":"§5.2, Proposition 5.2"},{"comment":"The approximation argument in the proof of Theorem 1.1 relies on applying the a priori estimates from Sections 5.3 and 5.4 to the regularized solutions u_i. For this to be valid, the constant A in (5.3) must be controlled uniformly in i; the text states that the constants in Corollary 5.4 and Proposition 5.6 are independent of i, which is correct because of (5.35). It would be helpful to state this uniformity explicitly at the point where A is introduced in the proof of Theorem 1.1, since the same A enters the exponent β in Proposition 5.1 and hence the final Hölder exponent α_o. As written, the reader has to assemble this from (5.35)-(5.38); the argument is sound but the presentation would be clearer if this uniformity were highlighted.","section":"§5.5, Proof of Theorem 1.1"}],"minor_comments":[{"comment":"There are typographical errors in the running text and headers, e.g. 'SCHAUDER ESTIMA TES' and 'A PPLICATIONS'. These should be corrected.","section":"Title and headings"},{"comment":"The proof uses 'since N≥2' to ensure that the chosen m satisfies m>3, which is needed for the quantitative estimate in Lemma 3.3. The proposition and the rest of the paper do not exclude N=1. Either state N≥2 as an assumption in this proposition or modify the choice of m so that the argument covers N=1 as well.","section":"§3.4, Proposition 3.8"},{"comment":"In the endpoint discussion, the expression q=N(p-1)/(N-p)_+ is undefined when N≤p. The convention for (N-p)_+ in the endpoint case should be made explicit, and the counterexample should be stated only in the range where N>p.","section":"§6.5"},{"comment":"The notation in the hypothesis 'R∈(0,1/2 R)' is circular and appears to be a typo; the intended condition is presumably R≤ρ_o/2 or a similar bound involving an auxiliary radius. Please rewrite the statement with unambiguous notation.","section":"§5.1, Proposition 5.1"}],"recommendation":"major_revision","confidential_remarks":"The decisive issue is whether the radius-free Campanato estimate asserted in Proposition 5.1 can be extracted from [11, Theorem 1.3] as claimed. If the authors can supply the missing proof, the rest of the argument appears structurally coherent. I would not reject on the basis of the current gap, but it is essential and cannot be fixed by a short sentence. The paper is long and partly overlaps with the authors' own preprint [8]; the overlap is acknowledged, but the dependence on [8] for the Harnack inequality and for the counterexamples should be kept in mind when assessing novelty."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe real news is that Theorem 1.1 is a genuine advance: local Hölder continuity of the spatial gradient for bounded weak solutions to parabolic p-Laplace systems with Hölder coefficients, for every p>1, including the sub-critical range 11 under a boundedness hypothesis, and the application to doubly nonlinear equations in the super-critical fast diffusion regime is structurally interesting.\n\nThe paper is carefully written and mostly self-contained. Section 3 develops gradient bounds for differentiable coefficients using energy estimates, Moser and De Giorgi iterations; Section 4 proves comparison estimates in detail; Section 5 builds the freezing argument. The authors are transparent about the main limitation: below p*, solutions need not be bounded (they cite their own preprint [8] for counterexamples), so Theorem 1.1 is only for bounded solutions. They also state explicitly that part of the manuscript is extracted from [8], and Section 6 depends on the unreviewed preprint [8] for boundedness and the time-insensitive Harnack inequality. These are disclosed dependencies, not hidden ones.\n\nThe soft spot is Proposition 5.1. It imports [11, Theorem 1.3] and asserts, without proof, that for coefficients b(t) depending only on time, the upper bound on the radius in [11] can be dropped. The stress-test note is right to flag this: the radius restriction might come from the parabolic sup-estimate and the intrinsic time scale, not from x-dependence. The identification (5.10) is algebraically correct, but the paper does not show why p close to 1 cannot reintroduce a radius bound. If (5.9) fails or constants degrade near the maximal radius, then the Campanato estimates (5.20)-(5.21), Proposition 5.6, and Theorem 1.1 are unproved. This is load-bearing, not a cosmetic gap.\n\nIf Proposition 5.1 holds, the paper closes a real gap and deserves to be a standard reference. But right now the freezing step rests on an unproved assertion. I would send it to peer review with a referee explicitly asked to verify Proposition 5.1 against [11], and I would not cite the theorem in my own work until that step is clarified.","headline":"Genuinely new Schauder result, but the freezing argument rests on an unproved radius-removal claim in Proposition 5.1 that should be checked before accepting.","tokens_in":77663,"tokens_out":3211,"would_cite":false,"duration_ms":33521,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K65","35K67","35B45","35B65","35K92","76S05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Bounded weak solutions to parabolic p-Laplace systems with Hölder spatial coefficients have locally Hölder gradients for every exponent p>1.","keywords":["Schauder estimates","parabolic p-Laplace systems","gradient Hölder regularity","bounded weak solutions","intrinsic cylinders","Campanato estimates","doubly nonlinear parabolic equations","fast diffusion"],"falsifier":"Take a frozen-coefficient system $\\partial_t w-\\operatorname{div}(b(t)(\\mu^2+|Dw|^2)^{(p-2)/2}Dw)=0$ with $b$ depending only on time and compute the mean oscillations of $Dw$ over nested intrinsic cylinders $Q_\\tau^{(\\lambda)}\\subset Q_R^{(\\lambda)}$ with $\\tau/R$ arbitrarily small, in dimensions $N\\ge 2$ and exponents $p\\le 2N/(N+2)$; a decay slower than $(\\tau/R)^{\\beta p}$ for some ratio outside the range permitted in [11] would falsify the radius-free version of Proposition 5.1 on which the main theorem rests.","tokens_in":76267,"feed_emoji":"📐","tokens_out":8883,"duration_ms":95201,"temperature":0.7,"pith_summary":"The paper establishes gradient Schauder estimates for bounded weak solutions of the parabolic $p$-Laplace system with a coefficient that is only Hölder continuous in the space variable. The main theorem states that the spatial gradient $Du$ is locally Hölder continuous in space and time for every $p>1$, with explicit quantitative bounds: on a compact set $K$, the supremum of $|Du|$ is controlled by the oscillation of $u$, the distance to the parabolic boundary, and the regularization parameter $\\mu$, and the Hölder modulus of $Du$ is given by a power of the intrinsic parabolic distance. This unifies gradient regularity theory that previously required the super-critical range $p>2N/(N+2)$, extends it to systems with coefficients, and covers the sub-critical range $11, p-Laplace gradients are Hölder","feed_subtitle":"Bounded solutions get Schauder estimates even below the critical exponent, unlocking doubly nonlinear equations.","key_machinery":"The central object is the intrinsic cylinder $Q_\\rho^{(\\lambda)}(z_o)=B_\\rho(x_o)\\times(t_o-\\lambda^{2-p}\\rho^2,t_o]$, whose time length is tuned to the gradient scale $\\lambda$. The proof freezes the coefficient $a(x,t)$ to $a(x_o,t)$, compares $u$ to the solution $w$ of the frozen-coefficient Cauchy-Dirichlet problem on such a cylinder, and controls the comparison by Lemma 4.7's estimate $\\int_{Q_R}|Du-Dw|^p\\,dxdt\\le CR^{\\alpha_*p}\\int_{Q_R}(\\mu^2+|Du|^2)^{p/2}\\,dxdt$. The decisive ingredient is the improved Campanato-type estimate (5.9), taken from [11], which gives decay of the mean oscillations of $Dw$ like $(\\tau/R)^{\\beta p}\\lambda^p$ on intrinsic cylinders of fixed geometry. Combining this decay with the quantitative gradient bounds and an interpolation in an intermediate radius yields the Campanato estimate (5.21) for $Du$, from which the Lebesgue representative of $Du$ is shown to be Hölder continuous.","core_discovery":"In the paper's own terms, the central discovery is Theorem 1.1: for $p>1$, $\\mu\\in[0,1]$, and a coefficient $a$ satisfying (1.2), every bounded weak solution has $Du\\in C^{\\alpha_o,\\alpha_o/2}_{\\mathrm{loc}}(E_T,\\mathbb{R}^{kN})$, with the quantitative local gradient bound (1.3) and the gradient Hölder estimate (1.4), where $\\alpha_o$ and $C$ depend only on $N,p,C_0,C_1,\\alpha,k$. The theorem is proved first under the a priori assumption that $Du$ is locally bounded and then by approximating the Hölder coefficient by smooth coefficients and passing to the limit, which also covers $\\mu=0$. Below the critical exponent $p_*=2N/(N+2)$, the paper notes that weak solutions need not be locally bounded, citing [8] for counterexamples, so the boundedness hypothesis is essential, not merely technical. Above $p_*$, the method also yields gradient bounds depending only on $L^p$-integrals of $Du$. Theorem 6.1 then converts these estimates, via a time-insensitive Harnack inequality, into gradient and gradient-Hölder estimates for doubly nonlinear equations in the range $01$, with the boundedness assumption in the sub-critical range and no boundedness assumption above $2N/(N+2)$.","The quantitative estimates in terms of oscillation give explicit control of $\\sup_K|Du|$ and of the Hölder modulus of $Du$ that is stable under approximation by smooth coefficients and persists in the limit $\\mu\\downarrow 0$.","For doubly nonlinear equations $\\partial_t u^q-\\operatorname{div}(|\\nabla u|^{p-2}\\nabla u)=0$ in the range $00$, with constants depending only on $N,p,q$.","The same methods give quantitative extinction-time decay estimates for fast-diffusion solutions, including oscillation bounds for $u$ and $\\nabla u$ near the extinction time.","The special case $p=2$ recovers the known gradient estimates for the singular porous medium equation in the range $(N-2)_+/N2N/(N+2)$; the present work simplifies its varying-cylinder geometry.","marker":"[6]"},{"why":"Supplies the counterexamples showing that sub-critical weak solutions need not be bounded and the time-insensitive Harnack inequality used for the doubly nonlinear application.","marker":"[8]"},{"why":"Recovered as a special case: the singular porous medium equation gradient estimates when $p=2$.","marker":"[21]"}],"fun_headline_variants":["Holder gradient bounds for all p>1 bounded p-Laplace systems","Bounded p-Laplace solutions have Holder gradients for all p>1","Holder regularity for p-Laplace gradients: all p>1, if bounded","Every p>1 p-Laplace system: Holder gradients for bounded solutions","p-Laplace gradient Holder: all p>1, bounded solutions only"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the improved Campanato estimate (5.9), imported without proof from [11] for coefficients depending only on time, remains valid on intrinsic cylinders of fixed geometry for all $p>1$ and all radius ratios once the radius restriction of [11] is dropped; if that imported estimate secretly requires a condition on $p$ or on the cylinder aspect ratio, Theorem 1.1 would lose its proof.","fun_headline_variants_meta":{"raw":{"variants":["Holder gradient bounds for all p>1 bounded p-Laplace systems","Bounded p-Laplace solutions have Holder gradients for all p>1","Holder regularity for p-Laplace gradients: all p>1, if bounded","Every p>1 p-Laplace system: Holder gradients for bounded solutions","p-Laplace gradient Holder: all p>1, bounded solutions only"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001351,"raw_usage":{"total_tokens":5511,"prompt_tokens":998,"completion_tokens":4513,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":614,"completion_tokens_details":{"reasoning_tokens":4417}},"tokens_in":614,"tokens_out":4513,"duration_ms":34897,"temperature":1.0,"reasoning_tokens":4417,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:27:49.619118+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a frozen-coefficient system $\\partial_t w-\\operatorname{div}(b(t)(\\mu^2+|Dw|^2)^{(p-2)/2}Dw)=0$ with $b$ depending only on time and compute the mean oscillations of $Dw$ over nested intrinsic cylinders $Q_\\tau^{(\\lambda)}\\subset Q_R^{(\\lambda)}$ with $\\tau/R$ arbitrarily small, in dimensions $N\\ge 2$ and exponents $p\\le 2N/(N+2)$; a decay slower than $(\\tau/R)^{\\beta p}$ for some ratio outside the range permitted in [11] would falsify the radius-free version of Proposition 5.1 on which the main theorem rests.","supporting_citations":[{"cited_title":"B¨ ogelein, F","cited_arxiv_id":null,"evidence_quote":"Provides the improved Campanato-type estimate for solutions with time-dependent coefficients that the freezing step imports as Proposition 5.1."},{"cited_title":"DiBenedetto, Degenerate parabolic equations","cited_arxiv_id":null,"evidence_quote":"Supplies the general Moser and De Giorgi iteration scheme, Sobolev embedding, and interpolation lemmas used for the quantitative gradient bounds."},{"cited_title":"Choe, H¨ older regularity for the gradient of solutions of certain singular parabolic systems","cited_arxiv_id":null,"evidence_quote":"Handles the sub-critical range in the scalar case and supplies the difference-quotient technique for $L^m$ gradient estimates."},{"cited_title":"B¨ ogelein and F","cited_arxiv_id":null,"evidence_quote":"Earlier freezing argument for variable-exponent systems with $p(x,t)>2N/(N+2)$; the present work simplifies its varying-cylinder geometry."},{"cited_title":"DiBenedetto, Y","cited_arxiv_id":null,"evidence_quote":"Recovered as a special case: the singular porous medium equation gradient estimates when $p=2$."}],"review_version":1}