{"id":"a2af8250-5edb-4c70-865c-8cbbc0e69564","arxiv_id":"2607.17912","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 Trudinger equation obey a quantitative sup-bound with optimal tail and a time-gapped strong Harnack inequality.","lead":"An analysis paper proves quantitative upper bounds and a strong parabolic Harnack inequality for the nonlocal Trudinger equation, a doubly nonlinear nonlocal diffusion equation with measurable kernels. If correct, it gives pointwise control of solutions by their past average and long-range tails, a central tool in nonlocal parabolic regularity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Caccioppoli inequality with time-dependent truncation (Prop 2.12) is the load-bearing unproved imported step; if the mollification argument fails for 1<p<2 the main theorems collapse.","rationale":"The reader's weakest-assumption pinpoints Proposition 2.12, and our independent inspection confirms that this is the single most load-bearing step. The time-dependent Caccioppoli inequality (2.4) is not proved in the manuscript; it is delegated to an unpublished/preprint reference for the fractional term and to exponential-mollification arguments for the time derivative. For p∈(1,2), the singular factor |u|^{p-2} makes the formal integration by parts (2.3) delicate, and without a rigorous derivation the sup-bound in Theorem 1.1 and hence the Harnack chain in Theorem 1.2 lose their foundation. I found no stronger internal inconsistency; the temporary L^{p-1+ε} assumption in Theorem 1.1 is secondary and partially addressed by Step 4's iteration. Because the gap is addressable by writing out the mollification argument or by weakening the theorem, the correct disposition remains conditional rather than acceptance or rejection. Hence the reader's verdict should not change.","tokens_in":38786,"tokens_out":4788,"duration_ms":46040,"concrete_test":"Reproduce the proof of Proposition 2.12 for 1<p<2 (e.g., p=3/2) with a nonconstant time-dependent level k(t), carrying out the exponential mollification in time explicitly. Track the boundary term at t0−τ and the term involving k'(t), and identify the minimal regularity needed to pass to the limit in the ∂t(|u|^{p-2}u) term. If the limit requires ∂t u ∈ L^1 or u ∈ L^{p-1+ε}_loc beyond Definition 2.9, then (2.4) is not established as stated; if the argument closes using only the assumptions of Definition 2.9, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strong Harnack inequality (Thm 1.2) rests on Theorem 1.1 and the measure-theoretic lemmas in Section 4, all of which use the Caccioppoli inequalities (2.4)–(2.6). Proposition 2.12, the time-dependent-level version (2.4), is the only step that produces the k'(t) terms needed to absorb the nonlocal tail in the proof of Theorem 1.1. However, its proof is not given: the fractional term is dismissed with a reference to [CCMV25, Lemma 3.23], and the ∂t(|u|^{p-2}u) integration by parts is justified only informally via 'exponential mollification' in [BDL21] and [Nak22a, Appendix A]. For 1<p<2, ∂t(|u|^{p-2}u) is singular and not well-defined for u merely in L∞(0,T;W^{s,p}) ∩ C([0,T];L^p) as required by Definition 2.9; identity (2.3) is formal. Since (2.4) is used with nonconstant k(t) in Theorem 1.1 (Step 1) and the simplified versions (2.5)–(2.6) are used throughout Section 4, a failure of this mollification argument would invalidate both main theorems. The companion preprint [CCMV25] is not made available in the manuscript, and the exponential-mollification references are not reproduced, so the gap is real and load-bearing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops pointwise regularity theory for the nonlocal Trudinger equation ∂t(|u|^{p-2}u)+Lu=0 with a measurable, bounded, symmetric singular kernel satisfying (1.2). It claims a quantitative L∞–Lν local bound with optimal tail (Theorem 1.1) and a strong parabolic Harnack inequality (Theorem 1.2), including the simplified sup u ≤ CH inf u for globally nonnegative weak solutions. The strategy is a DiBenedetto-type De Giorgi–Moser iteration: time-dependent Caccioppoli inequalities (Proposition 2.12), refined upper bounds, and measure-theoretic lemmas (De Giorgi type, expansion of positivity, weak Harnack) leading to the strong Harnack inequality. Much of the structure follows and adapts the authors' companion preprint [CN26].","tokens_in":39232,"tokens_out":10971,"duration_ms":102606,"significance":"If the result holds, it is a substantial advance: the strong parabolic Harnack inequality for the nonlocal Trudinger equation with measurable kernels and optimal tail conditions is new, going beyond the linear case of Kassmann–Weidner and the earlier weak Harnack or local-boundedness results. The paper contains a detailed iteration with explicit constants for many steps, and the statements are quantitatively precise. Its main weakness is that the proof is not fully self-contained: several load-bearing lemmas, especially the time-dependent Caccioppoli inequality and the measure-theoretic machinery, are delegated to companion preprints or to informal mollification arguments. I could not certify correctness of those steps from the manuscript alone.","major_comments":[{"comment":"This proposition is load-bearing: (2.4) is used with time-dependent k(t) in the proof of Theorem 1.1 (see the estimate of term IV around (3.10)–(3.15)), and the simplified versions (2.5)–(2.6) are used throughout Section 4. Its proof, however, is not given in the manuscript. The fractional part is dispatched to [CCMV25, Lemma 3.23], and the ∂t(|u|^{p−2}u) integration by parts is justified only informally through identity (2.3) and 'exponential mollification' cited from [BDL21] and [Nak22a, Appendix A]. For 1<p<2, u↦|u|^{p−2}u is singular, and a function in the class of Definition 2.9 does not have a time derivative in the usual sense; identity (2.3) is formal. Since both Theorem 1.1 and Theorem 1.2 lose their foundation if this mollification step fails, the authors should supply a complete proof of (2.3)–(2.4), or state and prove the exact regularization lemma under the hypotheses of Def","section":"§2.5, Proposition 2.12"},{"comment":"The measure-theoretic part is largely imported from the companion preprint [CN26]. Lemma 4.2 says 'crudely similar', Lemma 4.3 says 'completely same argument', Lemma 4.4 'almost repeated verbatim', Lemma 4.5 'identical', and Theorem 4.6 is stated without proof as a 'combination of Lemmas 4.4, 4.5 and Theorem 1.1'. Since [CN26] is not part of this submission, the proof chain from Caccioppoli to the weak Harnack inequality is not verifiable from the manuscript. This is not merely stylistic: Theorem 4.6 is the central ingredient in the proof of Theorem 1.2. The authors should include the missing arguments or make the companion preprint available and cross-checked in this submission.","section":"§4, Lemmas 4.2–4.5 and Theorem 4.6"},{"comment":"The crucial cancellation III2 + IV ≤ 0 depends on the choice of time-dependent level ℓ(t) and on the compressed estimate (3.15). For 1<p<2, the estimate of IV uses both k′(t)≥0 and u≤M0 on Qi, and the displayed chain contains multiple implicit constants and restrictions on k. Since this is the point where the time-dependent Caccioppoli inequality is actually used, I ask for a more detailed verification of the term IV and of the restrictions (3.18) and (3.24), so that the reader can check that the cancellation is legitimate for all p∈(1,∞).","section":"§3, Theorem 1.1 Step 1 (Eq. (3.15))"}],"minor_comments":[{"comment":"There are several typos and nonstandard locutions: 'at stake' should likely be 'in force' or 'in place'; 'we fave' in Lemma 4.2; 'Lebesgue instant' is nonstandard; 'the support department of Mathematics' is ungrammatical.","section":"Throughout"},{"comment":"The symbol w is used both for the auxiliary function and for the generic value u; this makes identity (2.3) hard to parse. Please use a consistent notation, e.g. u for the solution and v or τ for the integration variable.","section":"Equation (2.3)"},{"comment":"The remark says Lp-continuity follows from exponential mollification and cites footnotes of [MNY23], but Lp-continuity is already part of Definition 2.9. Please clarify whether it is an assumption or a consequence, and if a consequence, state it as a lemma.","section":"Remark 2.10"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's proof relies heavily on the authors' own companion preprint [CN26], and the key Caccioppoli step is delegated to informal mollification arguments. If the companion is under review elsewhere, the editor may wish to obtain a copy or ask the authors to include the necessary proofs. The central claims are plausible, but I could not certify the proof of Proposition 2.12 and the Section 4 chain from the present text."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nIf Theorem 1.2 is true, it is the first strong parabolic Harnack inequality for the nonlocal Trudinger equation with measurable kernels and quantitative tail control. That is a real step beyond [KW24] (linear p=2), [SZ25] (weak Harnack only), and [Nak22b] (mixed local/nonlocal). The authors also give a clean statement of the simpler sup ≤ C_H inf version for globally nonnegative solutions. The De Giorgi–Moser machinery is recognizable and mostly coherent; the tail terms in the estimates are explicit, and the paper does not obviously hand-wave the global-to-local tail absorption.\n\nThe weak spot is Proposition 2.12. This is the Caccioppoli inequality with time-dependent truncation k(t), and it is the exact step that produces the k'(t) terms used to absorb the nonlocal tail in the proof of Theorem 1.1. Its proof is not in the paper: the fractional term is deferred to [CCMV25, Lemma 3.23] and the ∂t(|u|^{p-2}u) integration by parts is justified only by invoking 'exponential mollification' in [BDL21] and [Nak22a, Appendix A]. For 1<p<2 that time derivative is singular and not well-defined under the minimal regularity in Definition 2.9; identity (2.3) is formal. Since (2.4) is then used with nonconstant k(t) in Theorem 1.1 Step 1, and the simplified versions (2.5)–(2.6) drive Section 4, a failure of that mollification argument would bring down both main theorems. I don't think the estimates themselves are wrong — this is standard machinery in the doubly nonlinear literature — but the authors need to either provide the mollification details or state Proposition 2.12 as a separate assumption. Right now the proof chain runs through an unreviewed companion preprint by the same authors, and that is a real burden even though self-citation alone is not a flaw.\n\nMinor: Proposition 3.1 assumes u∈L^{p-1+ε} and Step 4 removes it via a standard iteration, but the presentation is compressed. The proof of Theorem 4.6 is also delegated to 'combination' plus a reference to [CN26]. Theorems 1.1 and 1.2 are new, they are not restatements, and the structure is plausible. I would send this to a serious referee, with the explicit instruction to verify Proposition 2.12 or request a proof. If I needed the strong Harnack principle for a nonlocal doubly nonlinear problem, I would cite it — after checking that the companion preprint had appeared.","headline":"Genuinely new strong Harnack result for the nonlocal Trudinger class, but the proof's load-bearing Caccioppoli inequality with time-dependent levels is not actually proved here.","tokens_in":39632,"tokens_out":2465,"would_cite":true,"duration_ms":24697,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B65","35R09","47G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a quantitative strong parabolic Harnack inequality for nonnegative weak solutions of the nonlocal Trudinger equation, with optimal tail control.","keywords":["nonlocal Trudinger equation","parabolic Harnack inequality","doubly nonlinear equation","fractional Sobolev spaces","tail estimates","De Giorgi iteration","weak solutions","singular kernels"],"falsifier":"Find a measurable kernel satisfying (1.2) and a weak solution with 1<p<2 for which the formal identity ∂t g+(u,k(t)) used in Proposition 2.12 acquires a non-negligible correction term after mollification; or, more accessibly, verify whether Proposition 2.12 can be derived from Definition 2.9 without extra regularity. If such a counterexample exists, the asserted Caccioppoli inequality and hence Theorem 1.2 fail.","tokens_in":38683,"feed_emoji":"⚖️","tokens_out":4296,"duration_ms":40476,"temperature":0.7,"pith_summary":"This paper aims to establish the parabolic Harnack principle for the nonlocal Trudinger equation, a doubly nonlinear integro-differential equation whose operator has a measurable, bounded, symmetric singular kernel. The main theorem states that for nonnegative weak solutions in a large cylinder, the supremum of the solution over an earlier small cylinder, together with a time-averaged tail of its positive part, is controlled by the infimum over a later cylinder together with a time-averaged tail of its negative part; for globally nonnegative solutions this reduces to sup u ≤ C_H inf u. The paper also proves quantitative L∞-Lν local bounds with optimal tail terms. A sympathetic reader would care because Harnack inequalities are the sharpest pointwise regularity available for solutions with rough data, and this extends the known linear and p-Laplacian results to the full doubly nonlinear range with sharp tail dependence.","feed_headline":"Strong Harnack inequality proven for nonlocal Trudinger equation","feed_subtitle":"Local maxima of solutions are controlled by later minima plus a negative tail term, under optimal assumptions.","key_machinery":"The load-bearing object is the Caccioppoli inequality with time-dependent truncation levels (Proposition 2.12), which controls the energy g±(u,k(t)) = ±(p−1)∫_k^u |τ|^{p−2}(τ−k)± dτ, the fractional Gagliardo seminorm, and the nonlocal interaction term by boundary energy, cut-off errors, and a signed term involving k′(t). This inequality is applied with carefully designed moving levels k(t) that absorb positive tails, yielding refined sup-bounds. The measure-theoretic core consists of De Giorgi-type lemmas—critical mass, expansion of positivity, and a nonlocal measure-shrinking lemma—which lead to a weak Harnack inequality with positive exponent; Theorem 1.2 chains that weak Harnack inequalit","core_discovery":"The central claim, Theorem 1.2, is that every nonnegative weak solution of ∂t(|u|^{p−2}u) + Lu = 0 in a cylinder of radius R0 = 4·6^{1/(sp)}ρ satisfies a strong Harnack estimate: the supremum over B_ρ × (t0 − (2ρ)^{sp}/2, t0] plus a positive-tail term is bounded by a constant C_H times the infimum over a later cylinder B_ρ × (t0 + 3(4ρ)^{sp}/4, t0 + (4ρ)^{sp}] plus a negative-tail term of higher integrability. If u ≥ 0 globally, the inequality becomes the classical sup u ≤ C_H inf u. The proof chains a refined energy-based iteration with a weak Harnack inequality, under the optimal tail condition Tail(u; B_{R0}) ∈ L^{p−1+ε}_loc.","pith_inferences":["If the time-dependent Caccioppoli inequality can be justified without exponential mollification for singular kernels in 1<p<2, the same chaining argument would likely give local Hölder continuity for the nonlocal Trudinger equation; the paper does not state this explicitly.","The tail-dependent formulation suggests a testable extension to equations with drift terms: adding a first-order term should shift the chaining cylinders but preserve the sup-plus-tail versus inf-plus-tail structure.","For global weak solutions, the time-gap collapse noted in the paper hints that a time-insensitive Harnack estimate may hold in the whole-space setting, analogous to known local results; this is an extrapolation beyond the stated theorem.","A numerical check of the Caccioppoli inequality with a simple singular kernel (e.g., k(x,y)=|x−y|^{−d−sp}) in p∈(1,2) could expose whether the unproved integration-by-parts step produces hidden boundary terms; the paper does not perform such a check."],"forward_implications":["For globally nonnegative weak solutions, the strong Harnack inequality sup u ≤ C_H inf u holds, implying the usual parabolic Harnack principle.","The estimate is quantitative: constants depend only on d, s, p, Λ, and the tail terms are optimal in their scaling.","The time gap between the sup-cylinder and inf-cylinder is intrinsic to the nonlocal doubly nonlinear setting; the estimate fails to be time-insensitive for local solutions.","Together with the refined L∞-Lν bound (Theorem 1.1), the result yields pointwise control that is the best available regularity for these rough-coefficient equations.","The assumption Tail(u; B_{R0}) ∈ L^{p−1+ε}_loc is the optimal integrability condition under which the Harnack estimate holds."],"fun_headline_variants":["Sharp Harnack inequality for nonlocal Trudinger equation","Nonlocal Trudinger solutions obey strong Harnack estimates","Optimal tail conditions give Harnack for nonlocal Trudinger","Harnack bounds for doubly nonlinear nonlocal Trudinger"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire proof rests on the time-dependent Caccioppoli inequality (Proposition 2.12), whose integration by parts in time is not proved in the paper but delegated to exponential-mollification arguments in earlier references; if that step fails for singular measurable kernels in the range 1<p<2, both main theorems lose their foundation.","fun_headline_variants_meta":{"raw":{"variants":["Sharp Harnack inequality for nonlocal Trudinger equation","Nonlocal Trudinger solutions obey strong Harnack estimates","Optimal tail conditions give Harnack for nonlocal Trudinger","Harnack bounds for doubly nonlinear nonlocal Trudinger"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000549,"raw_usage":{"total_tokens":2411,"prompt_tokens":647,"completion_tokens":1764,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":391,"completion_tokens_details":{"reasoning_tokens":1691}},"tokens_in":391,"tokens_out":1764,"duration_ms":12365,"temperature":1.0,"reasoning_tokens":1691,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T16:37:51.822283+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a measurable kernel satisfying (1.2) and a weak solution with 1<p<2 for which the formal identity ∂t g+(u,k(t)) used in Proposition 2.12 acquires a non-negligible correction term after mollification; or, more accessibly, verify whether Proposition 2.12 can be derived from Definition 2.9 without extra regularity. If such a counterexample exists, the asserted Caccioppoli inequality and hence Theorem 1.2 fail.","supporting_citations":[],"review_version":1}