{"id":"51eb7f6f-9a78-46b8-ac6a-7dfd17000b44","arxiv_id":"2602.07647","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Singular fractional p-Laplacian diffusion satisfies local integral Harnack estimates that imply finite-time extinction with (T*-t)^{1/(2-p)} decay.","lead":"This paper proves integral Harnack inequalities for singular fractional p-Laplacian diffusion, showing how the local mass and peak of a solution compare across time. It then uses them to bound how quickly nonnegative solutions extinguish in finite time.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Theorem 1.2 asserts WLOG t1≤t2 for the sup/inf times; time reversal is not available for (1.2), so the L1-L1 estimate is not fully justified as written.","rationale":"The reader's weakest assumption was the reliance on [33] for the energy estimate and embedding, and the local-boundedness/subcritical assumptions. Those are legitimate but the paper explicitly adopts the solution class of [33], so that concern is at least partially addressed by the manuscript's own definitions. The time-ordering WLOG in the proof of Theorem 1.2 is an internal and central gap: it is not justified by any symmetry of the equation, and the main L1-L1 estimate — which feeds into Theorems 1.3 and 1.6 — depends on it. The paper also contains a separate inconsistency in Theorem 1.4 (the final exponent in the proof is 1/(2−p), while the statement has 1/r, and Lemma 5.1 has a sign slip), which the reader correctly noted; that error is peripheral to the extinction claims but reinforces that the manuscript needs careful revision. The most load-bearing issue for the advertised extinction estimate is the ordering gap in Theorem 1.2. Since a fix is likely available by an absolute-value estimate, the appropriate verdict remains conditional rather than reject: the argument should be repaired before the L1-L1 and consequent decay claims are accepted as proved.","tokens_in":47408,"tokens_out":44127,"duration_ms":386282,"concrete_test":"Re-derive (4.14) under the opposite ordering t2<t1. Write the weak formulation on [t2,t1] to obtain S_n ≤ J1 + |∫_{t2}^{t1}D|, where D is the diffusive integrand; then check whether the estimates (4.15)-(4.20) control |∫_0^t D| with the same constants and ε/b iteration. If they do, the WLOG can be replaced by an absolute-value argument and the proof closes. If they do not, construct a numerical solution (e.g. N=2, s=1/2, p=3/2) where the mass on B_ρ attains its maximum after the mass on B_{2ρ} attains its minimum, and verify whether (4.14) fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Theorem 1.2 (Section 4), for each n the authors choose t1 realizing sup_{0<τ<t} ∫_{B_n} u(·,τ) and t2 realizing inf_{0<τ<t} ∫_{B_{2ρ}} u(·,τ), then state: “Without loss of generality, we shall assume henceforth that 0≤t1≤t2≤t.” This is not a WLOG: the fractional p-Laplacian evolution is not invariant under t ↦ T−t, and if the local mass supremum occurs after the infimum, the test-function computation (4.14) integrates over a backward time interval. The subsequent bounds on J3, J4, J5 are upper bounds for the forward diffusive integral over (t1,t2); with reversed endpoints the boundary term changes sign and those estimates do not control the reversed integral unless one explicitly bounds |∫_{t2}^{t1}D| by the same J3+J4+J5 estimates. The manuscript does not supply that argument. This is load-bearing because Theorem 1.6(i) is derived by applying Theorem 1.2 (via Corollary 4.2) to the extinction interval; if the L1-L1 estimate is not established, the central mass-decay estimate is not established either. A repair likely exists — integrate over [min(t1,t2), max(t1,t2)] and bound the absolute value of the diffusive integral — but it is absent from the text.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies local weak solutions (Definition 2.1) of the singular fractional diffusion equation u_t + L_K u = 0, where 1<p<2, s∈(0,1), and K is a measurable symmetric kernel satisfying two-sided bounds comparable to |x-y|^{-(N+ps)}. The main results are integral Harnack-type estimates: an L^r-L^∞ estimate (Theorem 1.1), an L^1-L^1 estimate (Theorem 1.2), an L^1-L^∞ estimate under λ_1>0 (Theorem 1.3), and a backward L^r-L^r estimate (Theorem 1.4), all with explicit nonlocal tail terms. These are then applied to the Cauchy-Dirichlet problem: Theorem 1.5 gives a finite extinction time T_* with quantitative bounds in terms of initial data, and Theorem 1.6 gives decay rates for the local mass and supremum as t approaches T_*. The proofs combine energy estimates and a fractional parabolic embedding quoted from Liao [33], a De Giorgi-type iteration, and a time-mollification argument in Appendix A that is specifically designed to avoid assuming an integrable time derivative.","tokens_in":47754,"tokens_out":9835,"duration_ms":94252,"significance":"If the results are correct, they constitute a substantial extension of the DiBenedetto–Gianazza–Vespri Harnack machinery to singular nonlocal fractional p-Laplacian operators with measurable kernels, and they provide quantitative extinction rates without relying on comparison principles. The explicit tail terms and the careful time-mollification lemmas in Appendix A are notable technical contributions. The paper is clearly written and the overall strategy is plausible. However, several load-bearing proof issues must be fixed before the results can be considered established.","major_comments":[{"comment":"The statement 'Without loss of generality, we shall assume henceforth that 0≤t1≤t2≤t' is not a WLOG: equation (1.2) is not invariant under time reversal, and the test-function identity in Definition 2.1 is only applied on forward time intervals. If the time t1 realizing the supremum of ∫_{B_n}u occurs after the time t2 realizing the infimum over B∞, the boundary term in (4.14) changes sign and the subsequent estimates for J3, J4, J5 control the forward diffusive integral, not the reversed one. This gap is load-bearing because Theorem 1.6(i) is derived through Corollary 4.2 from Theorem 1.2. A repair likely exists by integrating over [min(t1,t2), max(t1,t2)] and bounding the signed diffusive integral by its absolute value using the same positive-part estimates, but the manuscript does not supply this argument.","section":"Section 4, proof of Theorem 1.2, Eq. (4.14)"},{"comment":"The final displayed inequality in the proof of Lemma 5.1 has a minus sign before the γ/σ^{N+ps} term, whereas the statement of the lemma and the preceding estimates (5.2), (5.3), (5.6) imply a plus sign. From 0 ≥ I_1 + I_2 and I_2 ≥ −G one obtains ∫_B u^r(t0) ≤ ∫_{B̂} u^r(0) + rG, not the printed inequality with minus. As written, the proof's concluding chain is internally inconsistent; the displayed sign should be corrected.","section":"Section 5, Lemma 5.1"},{"comment":"The statement of Theorem 1.4 gives the perturbation term as γ (t^r/ρ^{λ_r})^{1/r}, but the proof's iteration step (with Lemma 2.6 and η=(2−p)/r) yields exponent 1/(2−p) in the final bound. These exponents differ for p∈(1,2), so the theorem as printed is not the statement that is proved. The proof appears to establish the exponent 1/(2−p); please correct the statement or adjust the proof accordingly.","section":"Section 5, Theorem 1.4"}],"minor_comments":[{"comment":"Please add a sentence confirming that Propositions 2.3 and 2.4, quoted from [33], apply to local weak solutions in the sense of Definition 2.1 without any additional time-regularity assumption. The paper's stated novelty is avoiding an integrable time derivative, so the reader needs this compatibility stated explicitly.","section":"Section 2"},{"comment":"In the definition of A_τ the text reads 'u(x, τ)> u(x, τ)'; this should be 'u(x, τ)> u(y, τ)'.","section":"Section 4, Lemma 4.1"},{"comment":"The displayed integration bound contains ∫_0^1 1dτ; this should be ∫_0^t 1dτ.","section":"Section 6, proof of Theorem 1.5(i)"},{"comment":"The final condition on t appears as 't ≥ γ∗ + C1 ∥u0∥...'; based on the preceding computation this should be 't ≥ (γ∗/C1) ∥u0∥^{2−p}_{L^2(Ω)} |Ω|^{λ_2/(2N)}'.","section":"Section 6, proof of Theorem 1.5(ii)"},{"comment":"In the text 'Not that the estimates in (a), (b) above coincide' should read 'Note that'.","section":"Section 5, proof of Lemma 5.1"},{"comment":"The keyword phrase 'Harnack tipe inequality' contains a typo; it should be 'Harnack type inequality'.","section":"Abstract/Keywords"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the overall strategy is credible. The major issues identified affect the statement/proof of central theorems but appear repairable: the WLOG time-ordering in Theorem 1.2 needs an absolute-value argument, and the sign/exponent errors in Section 5 are local corrections. I therefore recommend major revision rather than rejection. No concerns about citation practices or novelty disclosure beyond the normal expectation that the authors verify the compatibility of the quoted estimates from [33] with their solution definition."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper proves L1-L1 and L1-L∞ Harnack inequalities for local weak solutions of u_t + L_K u = 0 with 1<p<2 and measurable symmetric kernel K, and then uses them to derive quantitative extinction rates for the Cauchy-Dirichlet problem. The results are new, the iteration machinery is standard but carefully adapted to the nonlocal setting, and the authors avoid the usual integrable-time-derivative assumption via a mollification appendix. The significance is real: these are the first integral Harnack estimates for this class, and the extinction rates for the fractional p-Laplacian do not rely on comparison principles.\n\nThe proof of Theorem 1.2 has a gap that the stress-test note correctly identifies. The authors choose t1 (argmax of B_n mass) and t2 (argmin of B_∞ mass) and assert WLOG t1≤t2. That is not a WLOG because the equation is not invariant under time reversal. If the max occurs after the min, the test-function computation integrates backward in time, and the resulting sign in front of the diffusive integral flips. The estimates that follow bound the forward diffusive integral from above; in the reversed case you need a lower bound (or an absolute value estimate) to get the same conclusion. The repair is straightforward — integrate over [min(t1,t2), max(t1,t2)] and bound the absolute value of the diffusive integral — but it is absent. Since Theorem 1.6(i) and Corollary 4.2 use Theorem 1.2, the extinction rate proof inherits this gap.\n\nThere are also two smaller display errors: Lemma 5.1's final inequality has a minus sign in front of the nonlocal term, while the preceding inequalities give a plus (the lemma statement is fine); and Theorem 1.4's statement writes (t^r/ρ^{λ_r})^{1/r} while the proof concludes with exponent 1/(2-p). The latter is likely a typo but should be fixed.\n\nThe references look appropriate, the proofs are long but mostly checkable, and the authors are careful about the solution notion. The ellipticity of the energy estimates is quoted from [33]; that is fine, though it means the Harnack estimates inherit whatever regularity hypotheses [33] carries.\n\nWho should read this: people working on nonlocal parabolic regularity, initial traces, and extinction phenomena. It deserves peer review — the gap is fixable and the results are worth having. I would not cite it in its current form, but with the time-order issue repaired it would be a solid reference.\n\nRecommendation: send it to a strong referee, and ask specifically for a check of Theorem 1.2's time ordering and the displayed exponent/sign issues in Theorems 1.4 and Lemma 5.1.\n\nBest.","headline":"New L1-L1 Harnack estimates and extinction rates for singular fractional p-Laplacian; the main theorem's proof has a time-ordered gap that needs a fix, but the core program looks sound.","tokens_in":48253,"tokens_out":12036,"would_cite":false,"duration_ms":96094,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K67","35B65","35K92","35Q35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that local weak solutions of singular fractional p-Laplacian diffusion obey integral Harnack inequalities and that nonnegative solutions to the Cauchy-Dirichlet problem extinguish in finite time with explicit decay rates.","keywords":["fractional p-Laplacian","singular diffusion","Harnack-type inequality","L1-L1 Harnack estimate","extinction time","decay rate","nonlocal parabolic equations","measurable kernels"],"falsifier":"Search for a measurable-kernel solution of (1.2) with p<2 and no integrable time derivative for which the quoted energy estimate (Proposition 2.4) fails, or run a high-resolution numerical scheme for the Cauchy-Dirichlet problem with p=1.5 and a discontinuous kernel and check whether the local mass obeys ((T*-t)/ρ^{λ1})^{1/(2-p)}; a counterexample to either would invalidate the central claim.","tokens_in":47278,"feed_emoji":"⏳","tokens_out":6384,"duration_ms":60273,"temperature":0.7,"pith_summary":"The paper targets the singular range 1<p<2 of fractional p-Laplacian-type diffusion, where the operator degenerates when nearby values coincide, a regime that models pseudoplastic fluids and long-range nonlocal interactions. It establishes L1-L1 and L1-L∞ Harnack-type inequalities for local weak solutions with measurable, bounded, symmetric kernels, and it deliberately avoids assuming an integrable time derivative. Applying these estimates to the Cauchy-Dirichlet problem shows that every nonnegative bounded weak solution vanishes after a finite time T*, with quantitative decay of the local mass and of the supremum as t approaches T*. If correct, the results provide a flexible tool for transferring measure information in time and for computing extinction profiles in nonlocal parabolic problems.","feed_headline":"Harnack bounds pin down extinction rates for singular diffusion","feed_subtitle":"New integral Harnack inequalities give finite extinction time and explicit decay rates.","key_machinery":"The core mechanism is an iterative De Giorgi-type energy argument combined with a nonlocal tail functional that measures long-range mass. The paper relies on a parabolic fractional Sobolev embedding and an energy estimate for solutions, quoted from prior work, and on a time-mollification procedure that allows admissible test functions even when solutions lack an integrable time derivative. The exponent λ1 = N(p-2)+ps controls the decay rates, while the tail term carries the far-field information needed for local statements.","core_discovery":"The central claim is that a locally bounded nonnegative local weak solution of u_t + L_K u = 0 satisfies the L1-L1 Harnack estimate: the supremum over time of the mass in a ball is controlled by the infimum over time of the mass in a larger ball, plus tail terms that capture long-range spatial influence. Chaining this with an Lr-L∞ estimate yields an L1-L∞ estimate in the supercritical range λ1 > 0. For the Cauchy-Dirichlet problem the same machinery, using time-mollified test functions, proves finite extinction: the solution is identically zero after a time T*, and the local mass obeys ∫_{Bρ} u(x,t)dx ≤ γ ((T*-t)/ρ^{λ1})^{1/(2-p)}, with a comparable sup-norm decay when λ1 > 0.","pith_inferences":["If the quoted energy estimate holds for rougher solution classes, the same iteration should yield Harnack and extinction estimates for very weak or measure-valued solutions, extending the notion of solution.","The exponent 1/(2-p) is the fractional analogue of the fast p-Laplacian extinction exponent; numerical experiments for p<2 with discontinuous kernels could test whether this rate is optimal.","The L1-L1 estimate could be iterated with the fractional Sobolev embedding to produce a shorter route to Hölder continuity than existing oscillation arguments.","The tail terms suggest that in unbounded domains the decay rate depends on the far-field profile; checking whether a shrinking-tail condition recovers the global whole-space rates would clarify the role of the tail."],"forward_implications":["Every nonnegative bounded weak solution of the Cauchy-Dirichlet problem has a finite extinction time T* bounded by constants times powers of the initial norm, after which the solution is identically zero.","The local mass decays as ((T*-t)/ρ^{λ1})^{1/(2-p)} for all 1<p<2, uniformly up to the extinction time.","In the range 2N/(N+s)<p<2, the same estimates give a sup-norm decay rate with explicit tail corrections.","The L1-L1 Harnack inequality transfers mass bounds uniformly in time, making it a direct tool for Hölder continuity and initial-trace arguments.","All conclusions hold for measurable, bounded, symmetric kernels, not just the prototype fractional p-Laplacian, so they apply to anisotropic nonlocal media."],"fun_headline_variants":["Harnack inequalities reveal extinction rates for fractional diffusion","Finite extinction time proven for singular fractional diffusion","Extinction rate of singular diffusion pinned by Harnack estimates","Harnack bounds give explicit decay rates near extinction","Integral Harnack estimates control fractional diffusion extinction"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The Harnack and extinction arguments depend on a quoted energy estimate and a fractional embedding that may require more time-regularity than the weak solutions of Definition 2.1 possess; if those estimates fail for measurable-kernel solutions without an integrable time derivative, the iteration collapses.","fun_headline_variants_meta":{"raw":{"variants":["Harnack inequalities reveal extinction rates for fractional diffusion","Finite extinction time proven for singular fractional diffusion","Extinction rate of singular diffusion pinned by Harnack estimates","Harnack bounds give explicit decay rates near extinction","Integral Harnack estimates control fractional diffusion extinction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000561,"raw_usage":{"total_tokens":2446,"prompt_tokens":636,"completion_tokens":1810,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":380,"completion_tokens_details":{"reasoning_tokens":1747}},"tokens_in":380,"tokens_out":1810,"duration_ms":11779,"temperature":1.0,"reasoning_tokens":1747,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T03:32:04.989880+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search for a measurable-kernel solution of (1.2) with p<2 and no integrable time derivative for which the quoted energy estimate (Proposition 2.4) fails, or run a high-resolution numerical scheme for the Cauchy-Dirichlet problem with p=1.5 and a discontinuous kernel and check whether the local mass obeys ((T*-t)/ρ^{λ1})^{1/(2-p)}; a counterexample to either would invalidate the central claim.","supporting_citations":[],"review_version":1}