{"id":"75496c25-b4cf-47bc-bf3b-d5a14e50b5f1","arxiv_id":"2412.03770","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Subsolutions to parabolic fractional p-Laplacian equations are locally bounded for all p>1, with a bound using an L1-in-time nonlocal tail.","lead":"This paper proves a quantitative local boundedness estimate for subsolutions of a class of nonlinear nonlocal parabolic equations, the fractional p-Laplacian type, valid for every p>1. It extends a known linear-case result to the nonlinear setting with an improved nonlocal tail term using only an L1 average in time.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.3 states only pointwise subsolution, but the proof requires u_+^{2p−2} (p≥2) or u_+^{1+ξ} (p<2) and finite tail; these are not implied by the natural energy class, so the advertised local boundedness of weak subsolutions is not established.","rationale":"I read the paper as establishing a De Giorgi-type conditional estimate, and the main algebraic structure of the iteration appears coherent. The load-bearing weakness is the mismatch between the theorem's hypothesis and the integrability actually used in the proof. The reader's weakest_assumption identifies this same gap: for p≥2, u_+∈L^{2p−2} is not automatic from L^p(I;W^{s,p})∩L∞(I;L^2), and for p<2, the iteration needs u_+∈L^{1+ξ} with ξ above the stated threshold. I agree with this assessment. The coefficient typo in Proposition 2.5 Case 2 is a separate, easily repairable issue that would break the application of Lemma 2.2 if left uncorrected. The boundedness of f in Lemma 2.2 is another standard but omitted truncation step. None of these points disproves the conditional estimate; they show that the paper's advertised claim for all energy-class subsolutions needs an explicit higher-integrability hypothesis or an additional bootstrap argument. The reader's CONDITIONAL verdict is therefore appropriate, and I would not change it.","tokens_in":31500,"tokens_out":22858,"duration_ms":228548,"concrete_test":"Fix p=4, d=1, s=0.1 and scan §3 to determine the maximal exponent q for which ∬v_l^q is controlled by Lemma 2.4(i) together with Proposition 2.6. If q<2p−2=6 for this parameter choice, then no argument in the proof supplies the finiteness of the RHS of Theorem 1.3(i), so the theorem must either state L^{2p−2} as an explicit hypothesis or prove it. As an additional probe, take a nonnegative function in L^p(B_1)∩L^2(B_1) but not L^{2p−2}(B_1), such as |x|^{-1/8} with p=4, and check whether it can be a local subsolution of the fractional p-Laplacian equation; if it can, the theorem as stated is vacuous for that case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.3 is stated for a subsolution satisfying only the pointwise inequality (1.3), but the proof uses additional integrability as an input. For p≥2, the final step (3.3) applies Proposition 2.5 to u_+, which requires u_+∈L^{2p−2}(I_R^⊖×B_R); for p∈(1,2), the iteration starts with A_0=∬u_+^{1+ξ} for ξ>d(2−p)/(sp), and Proposition 2.6 requires the energy of v_l^{(p−1+ξ)/p}. These integrability conditions do not follow from the natural energy space L^p(I;W^{s,p})∩L∞(I;L^2). For instance, with p=4, d=1, s=0.1, Lemma 2.4(i) gives at best u_+∈L^{4.8}, while 2p−2=6. Thus Theorem 1.3 is a conditional estimate: if the displayed norms are infinite, the inequality is vacuous; if they are finite, boundedness follows. The paper does not prove these norms are finite for the class of weak subsolutions it claims to treat, so the advertised local boundedness of local weak solutions is not established. A secondary gap is that Lemma 2.2 is applied to f(r)=sup_t∮v^p without first proving f is bounded; the standard truncation argument is not mentioned.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a quantitative local boundedness estimate for subsolutions of the parabolic equation ∂_t u − L_t u = 0, where L_t is a fractional p-Laplacian type operator with kernel comparable to |x−y|^{−d−sp}, for every p∈(1,∞). The estimate controls sup_{B_{R/2}×I_{R/2}^⊖} u by local averages of powers of u_+ and by a nonlocal parabolic tail with L^1-in-time weight. The proof combines a new Caccioppoli-type inequality (Lemma 2.1), a tail-cancellation mechanism (Proposition 2.6), and a De Giorgi iteration with carefully chosen level sets. The paper claims to extend the linear p=2 result of Kassmann and Weidner to all p>1.","tokens_in":31779,"tokens_out":11754,"duration_ms":124886,"significance":"If the estimate holds for the intended weak-solution class, this is a valuable contribution: it unifies the p=2 linear theory with the nonlinear fractional p-Laplacian case and improves earlier results by using an L^1-in-time tail rather than an L^∞-in-time tail. The proof is coherent and genuinely structural: the constants are explicit and no parameter is fitted, and Proposition 2.6's cancellation of the outer tail is an elegant and checkable step. The main deficit is that the theorem statement does not specify the integrability and regularity hypotheses that the proof actually uses, so the advertised local boundedness result is only conditional as stated.","major_comments":[{"comment":"The theorem as stated is broader than what the proof establishes. The proof applies Proposition 2.5 to u_+ at (3.3), which requires u_+ ∈ L^{2p−2}(I_R^⊖×B_R) when p≥2. For p∈(1,2), the iteration starts from A_0 = ∬ u_+^{1+ξ} with ξ > d(2−p)/(sp), and Proposition 2.6 and the subsequent estimates require this integral and Tail_{p−1}(u) to be finite. None of these conditions is stated in Theorem 1.3, and none follows from the pointwise inequality (1.3) or from the natural energy class L^p(I;W^{s,p})∩L^∞(I;L^2). For example, when p=4, d=1, s=0.1, Lemma 2.4(i) gives at most u_+∈L^{4.8} from the energy class, whereas 2p−2=6 is needed. If the displayed averages are infinite, the estimate is vacuous and no boundedness follows. To make Theorem 1.3 a local boundedness statement for weak subsolutions, the authors must either prove these integrability properties for their solution class or state them explicitly as hypotheses; they should also specify the global spatial class needed to define the tail.","section":"§1.2, Theorem 1.3; §3, (3.3) and (3.4)"},{"comment":"In the proof of Proposition 2.5, after the case sp≤1 is treated, the displayed iteration inequality reads f(r) ≤ (s−r)^{−1}(A_1+A_2) + (s−r)^{−p(d+sp)}B + [p/(p−1)] f((r+s)/2). Lemma 2.2 is then invoked, but it requires a contraction coefficient κ<1, whereas p/(p−1)>1 for p>1. The preceding calculation gives (p−1)/p as that coefficient, so this appears to be a typographical error; if the displayed coefficient is literal, the iteration argument for sp≤1 is invalid. The authors should correct the coefficient and re-verify that Lemma 2.2 applies in both cases.","section":"§2, Proposition 2.5, Case 2 (sp≤1)"}],"minor_comments":[{"comment":"Lemma 2.2 is applied to f(r)=sup_{I_r^⊖} ∮_{B_r} v(t,x)^p dx without first proving that f is bounded on [R/2,R]. The standard truncation f_N=min(f,N), followed by N→∞, supplies the missing justification and should be mentioned.","section":"§2, Proposition 2.5"},{"comment":"The paper defines subsolutions through the pointwise inequality (1.3) but all later calculations, especially the integration by parts in time in Lemma 2.1, are formal. The authors should state the weak/energy formulation of (1.1) and the class of admissible test functions for which Lemma 2.1 is proved.","section":"§1.1.1 and §2, Lemma 2.1"},{"comment":"There are several typographical slips: 'supsolution' in §1.1.1, 'For the safe of completeness' at the start of Part (2) of the proof of Theorem 1.3, and inconsistent notation such as 'p−1/p' where '(p−1)/p' is meant. These should be corrected in a revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern lands: the paper's central estimate is sound as an a priori estimate, but Theorem 1.3 as stated does not prove local boundedness for the weak-solution class it advertises because the required integrability and tail finiteness are not imposed or derived. This is fixable by adding explicit hypotheses or by proving the needed higher integrability, so I do not recommend rejection. The coefficient issue in Proposition 2.5, if not a typo, would break the sp≤1 case and must be fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper does what it claims, but the headline theorem overclaims. The De Giorgi iteration is coherent and the new Caccioppoli inequality with the free parameter ξ is a genuine advance, letting the tail use only L1-in-time for all p>1. That matches the linear p=2 theory and improves [1], [9], [15]. If the paper were just a refinement of known tools, I'd call it routine; the level-set truncation in Prop 2.6 that cancels the intermediate tail is the clever part, and it seems to work.\n\nSoft spots in order of importance. First, Theorem 1.3 states only that u is a subsolution in the pointwise sense. The proof of the p≥2 case applies Prop 2.5 to u_+ and needs u_+ in L^{2p−2} locally; the p<2 case needs u_+ in L^{1+ξ} with ξ>d(2−p)/(sp). Those are not consequences of the natural energy space L^p(I;W^{s,p})∩L∞(I;L^2). So as stated, the theorem is a conditional estimate: if the displayed right-hand side is infinite, it says nothing, and the paper does not show these norms are finite for the class it claims to cover. This is fixable — state the theorem with explicit integrability hypotheses, or add a preliminary step proving them under the energy-class assumption. But as written it is a real gap in the statement, not just a cosmetic one.\n\nSecond, Prop 2.5, Case 2 has a typo: the coefficient p/(p−1) in the iteration inequality should be (p−1)/p. Otherwise κ>1 and the iteration doesn't close. A referee will catch it, but it should be fixed. Third, minor: Lemma 2.2 is applied to f(r)=sup_t ∮ v^p without proving f bounded; the standard truncation argument is omitted. Also two references ([2] and [8]) are never cited.\n\nI did not re-derive the Sobolev embeddings borrowed from [9], but they look standard and the iteration algebra checks out. The core method is sound.\n\nWho this is for: people working on nonlocal parabolic regularity. It deserves a serious referee; the contribution is real even though the presentation needs work. My recommendation: send to peer review with a request to tighten the statement and fix the typo.","headline":"Real improvement in local boundedness for fractional p-Laplacian parabolic equations, but the main theorem is stated too loosely: the proof needs extra integrability that isn't in the hypotheses.","tokens_in":32402,"tokens_out":3248,"would_cite":true,"duration_ms":32660,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B65","35R11","35K55","60J75"],"pacs":[],"model":"deepseek-v4-flash","headline":"Subsolutions of fractional p-Laplacian parabolic equations are locally bounded for every p>1, with a bound that matches the linear case p=2.","keywords":["fractional p-Laplacian","local boundedness","parabolic equations","De Giorgi-Nash-Moser iteration","Caccioppoli inequality","nonlocal parabolic tail","subsolution","fractional Sobolev spaces"],"falsifier":"Take p≥2 and construct a function u on a cylinder satisfying the weak subsolution inequality (1.3) with finite local means of u_+^{2p−2}, u_+^p, and finite tail Tail_{p−1}, but with sup of u over B_{R/2}×I_{R/2}^⊖ infinite. Theorem 1.3 would be false. A concrete candidate would be a solution with a growing singularity near the boundary of the smaller cylinder; numerical or analytic construction of such an example would settle the claim.","tokens_in":31177,"feed_emoji":"📐","tokens_out":4529,"duration_ms":42030,"temperature":0.7,"pith_summary":"The paper proves that every local subsolution of a parabolic equation driven by a fractional p-Laplacian type operator is locally bounded above, for every p in (1,∞). This fills a gap: prior results either handled only p≥2, or used a parabolic tail with an L∞-norm in time that never degenerates to the linear p=2 estimate. The new bound is explicit, with the local supremum controlled by local mean values of the positive part and a single nonlocal parabolic tail that uses only the L1-norm in time. A reader should care because this is the missing boundedness step needed to build Hölder regularity and Harnack estimates for these nonlinear nonlocal parabolic equations.","feed_headline":"Nonlocal parabolic subsolutions stay locally bounded for all p>1","feed_subtitle":"A new estimate controls the supremum by an L1-in-time tail, matching the linear p=2 case.","key_machinery":"The proof is carried by a Caccioppoli-type inequality (Lemma 2.1) for the truncated function v=(u−k)_+ raised to a power w=$v^{{(p−1+ξ)/p}}$, together with a decomposition of the nonlocal supremum tail into an intermediate tail, controlled by the equation, and a remaining tail, controlled by cancellation inside the iteration. A De Giorgi–Nash–Moser iteration then runs in two regimes, p≥2 and p∈(1,2), using fractional Sobolev inequalities with different exponents.","core_discovery":"Theorem 1.3 is the central claim: if u satisfies ∂_t u − L_t u ≤ 0 on a backward cylinder, then sup over the half-size cylinder is at most constants times sums of local means of powers of u_+ (2p−2 and p for p≥2; 2p−2, p, and 1+ξ for p∈(1,2)) plus the parabolic tail Tail_{p−1}, defined with the L1-norm in time. The statement is specifically designed so that when p=2 it reproduces the known linear estimate exactly, without the extra constant term that appears in earlier fractional p-Laplacian results.","pith_inferences":["Not in the paper: a natural next step is to use Theorem 1.3 in a Moser or Krylov–Safonov iteration to obtain Hölder continuity for all p>1; the linear case p=2 already has such a theory, and the boundedness proved here is the missing input.","Not in the paper: the time-dependent level k(t) chosen to absorb the nonlocal tail suggests the same truncation could handle kernels comparable to |x−y|^{−d−sp} with slowly varying coefficients, such as variable-order s(x,y), where an L1-tail would be harder to define.","Not in the paper: one could test whether the exponent 2p−2 in the p≥2 bound is necessary by constructing extremal functions where the u_+^p term is small and the supremum is controlled by the tail; the paper does not address optimality."],"forward_implications":["If u is a subsolution to ∂_t u − L_t u = f with f ∈ L^1_{t,x}, then the transformation v(t,x)=u(t,x)−∫_{t0}^t ‖f(s,·)‖_{L∞(B_R)} ds reduces to the homogeneous case, so boundedness transfers to equations with source terms.","For p=2 the estimate collapses to the linear bound with (∫∫ u_+^2)^{1/2} plus Tail_1, matching the known linear parabolic result.","For p∈(1,2), choosing the allowed ξ yields an explicit finite bound, so local weak subsolutions in the stated integrability class are locally bounded for every p>1, including the previously restricted range.","The L1-in-time parabolic tail replaces the L∞-in-time tail used in earlier works, which is the natural weakening needed to align nonlinear theory with the linear case.","The paper positions boundedness as a basis for later Hölder continuity and Harnack estimates for these equations."],"supporting_citations":[{"why":"Supplies the linear p=2 model and the L1-in-time parabolic tail strategy that the paper extends to all p>1.","marker":"[12]"},{"why":"Provides the prior local boundedness result for p≥2 and the Caccioppoli-type inequality that Lemma 2.1 adapts.","marker":"[15]"},{"why":"Gives a Hölder estimate with an optimal tail for nonlocal parabolic p-Laplace equations, serving as a comparison baseline.","marker":"[1]"},{"why":"Establishes local boundedness and Hölder continuity for parabolic fractional p-Laplace equations with restrictions, which the present theorem improves for all p>1.","marker":"[9]"},{"why":"Supplies the fractional p-Laplacian variational framework and the elementary inequality its Lemma 3.1, used inside the proof of Lemma 2.1.","marker":"[6]"},{"why":"Provides the iteration lemma (Lemma 7.1) used to force the De Giorgi–Nash–Moser sequence to zero.","marker":"[11]"}],"fun_headline_variants":["Fractional p-parabolic solutions: local boundedness for all p>1","New De Giorgi iteration extends local bounds to any p>1 for nonlocal parabolic equations","Nonlocal parabolic p-Laplacian: local bounds for every p>1 via L1-in-time tail","Local boundedness proven for all p>1 in fractional p-Laplacian parabolic equations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The estimate is meaningful only if the right-hand side quantities are finite: u_+ must lie in the stated Lebesgue classes ($L^{{2p−2}}$ for p≥2, $L^{{1+ξ}}$ with ξ>d(2−p)/(sp) for p<2) and the parabolic tail must be finite; the theorem's stated hypothesis, the pointwise inequality (1.3), does not by itself guarantee this.","fun_headline_variants_meta":{"raw":{"variants":["Fractional p-parabolic solutions: local boundedness for all p>1","New De Giorgi iteration extends local bounds to any p>1 for nonlocal parabolic equations","Nonlocal parabolic p-Laplacian: local bounds for every p>1 via L1-in-time tail","Local boundedness proven for all p>1 in fractional p-Laplacian parabolic equations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001098,"raw_usage":{"total_tokens":4595,"prompt_tokens":971,"completion_tokens":3624,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":587,"completion_tokens_details":{"reasoning_tokens":3530}},"tokens_in":587,"tokens_out":3624,"duration_ms":26706,"temperature":1.0,"reasoning_tokens":3530,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:08:28.161664+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take p≥2 and construct a function u on a cylinder satisfying the weak subsolution inequality (1.3) with finite local means of u_+^{2p−2}, u_+^p, and finite tail Tail_{p−1}, but with sup of u over B_{R/2}×I_{R/2}^⊖ infinite. Theorem 1.3 would be false. A concrete candidate would be a solution with a growing singularity near the boundary of the smaller cylinder; numerical or analytic construction of such an example would settle the claim.","supporting_citations":[{"cited_title":"Differential Equations , 266 (2019), 7948–7979","cited_arxiv_id":null,"evidence_quote":"Provides the prior local boundedness result for p≥2 and the Caccioppoli-type inequality that Lemma 2.1 adapts."},{"cited_title":"and Kim, K.: A Hölder estimate with an optimal t ail for nonlocal parabolic p-Laplace equations, Annali di Matematica, 203 (2024), 109–147","cited_arxiv_id":null,"evidence_quote":"Gives a Hölder estimate with an optimal tail for nonlocal parabolic p-Laplace equations, serving as a comparison baseline."},{"cited_title":"and Zhou, S.L.: Local boundedness a nd Hölder continuity for the parabolic fractional p-Laplace equations, Calc","cited_arxiv_id":null,"evidence_quote":"Establishes local boundedness and Hölder continuity for parabolic fractional p-Laplace equations with restrictions, which the present theorem improves for all p>1."},{"cited_title":"and Palatucci, G.: Local behavio r of fractional p-minimizers, Ann","cited_arxiv_id":null,"evidence_quote":"Supplies the fractional p-Laplacian variational framework and the elementary inequality its Lemma 3.1, used inside the proof of Lemma 2.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the iteration lemma (Lemma 7.1) used to force the De Giorgi–Nash–Moser sequence to zero."}],"review_version":1}