{"id":"8a554d32-913a-4071-9730-2c4946e4b5f3","arxiv_id":"2506.07221","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Gradient estimates for solutions of ∂_t u = Δ_p u^q are proved on manifolds with Ricci curvature bounded below, in both slow and fast diffusion regimes, under a uniform bound on |∇v|^{p-2} v.","lead":"This paper proves Li-Yau type gradient estimates for positive solutions of the doubly nonlinear Leibenson equation on Riemannian manifolds whose Ricci curvature is bounded below by a negative constant. It covers both slow and fast diffusion regimes and works on noncompact manifolds, where previously only special cases such as the porous medium equation were understood.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.1 implicitly assumes f = Fα - φ is nonnegative, using real powers f^λ and dropping 2c1fφ; this fails for admissible p=2 solutions, so the Moser iteration does not justify the theorem.","rationale":"The paper's central claim is a Li-Yau type estimate for the doubly nonlinear equation, proved via Moser iteration on f = Fα - φ. The most load-bearing weakness is not the assumed lower bound Λmin (which is part of the theorem's hypotheses and does not need to be relaxed for the proof to function), but the unstated and false assumption that f ≥ 0. The proof of Lemma 2.2 discards 2c1fφ using '2c1 f φ ≥ 0', and Lemma 2.4 uses test functions and norms involving f^λ for real λ; both require nonnegativity. The theorem's assumptions do not imply this, and for p=2 the condition (1.6) is just Λmin ≤ v ≤ Λmax, so ∇v=0 is allowed. An explicit p=2 solution of the pressure equation with v(0,x)=1+|x|^2 gives Fα < 0 at x=0, hence f < 0. Therefore the Caccioppoli inequality (2.26), the mean value inequality (2.35), and Lemma 2.8 are not established for the actual f, and the sup bound for Fα in Theorem 2.9 does not follow from the written argument. This is a genuine gap, though likely repairable by replacing f with f_+ = max(f,0) in the Moser iteration, as is standard in the literature; hence the CONDITIONAL verdict is appropriate. The reader's weakest_assumption focused on Λmin denominators, but the reader's rationale independently flags the sign issue; my analysis agrees with that part of the rationale while identifying it as the primary obstruction. A secondary inconsistency in the KΛmax coefficient appears across (1.8), (2.25), and (2.39), but it does not affect the main logical structure.","tokens_in":17460,"tokens_out":15424,"duration_ms":156937,"concrete_test":"Compute f = Fα - φ for the p=2, q>1 pressure solution with v(0,x)=1+|x|^2 on ℝ^n. At x=0, verify ∇v=0 and v_t = 2n(q-1) > 0, so Fα = -α v_t/v < 0 while φ≥0; if f<0, the implicit nonnegativity assumption in Lemma 2.2 is false and the real-power Moser iteration in §2.4 is invalid for the stated theorem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 2.2 and the Moser iteration that follows rely on f = Fα - φ being nonnegative. The proof drops the term 2 c1 f φ using '2c1 f φ ≥ 0' (p.6), and Lemma 2.4 multiplies (2.23) by ψ = f^{λ-1}η^2 and forms f^λ, f^{λ+1} for real λ (p.7-8). No pointwise lower bound for f is proved, and the theorem's assumptions do not supply one. For p=2, q>1, (1.6) reduces to Λmin ≤ v ≤ Λmax, which allows ∇v=0. In the pressure formulation v_t = δ v Δv + |∇v|^2, take v(0,x)=1+|x|^2. At x=0, ∇v=0, v_t=2nδ>0, so Fα = -α v_t/v = -2nαδ <0; with φ≥0, f<0. This is a smooth positive local solution satisfying the theorem's hypotheses, so the sign assumption is not merely unproved but false. Hence the Caccioppoli inequality (2.26), the mean value inequality (2.35), and Lemma 2.8 do not apply to the actual f, and the derivation of Theorem 2.9 has a gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Li-Yau-type gradient estimates for positive smooth solutions of the doubly nonlinear equation ∂_t u = Δ_p u^q on complete Riemannian manifolds with Ricci curvature bounded below by a non-positive constant. In the slow-diffusion regime q(p-1)>1 and the fast-diffusion regime q(p-1)<1, the author introduces a pressure variable v, assumes the two-sided bound Λmin ≤ |∇v|^{p-2}v ≤ Λmax, and uses a Bochner formula, an auxiliary function φ, a Caccioppoli inequality, and Moser iteration to bound the quantity Fα = |∇v|^p/v ∓ α(∂_t v)/v in geodesic cylinders and then globally as R→∞. The main results are Theorem 1.1 (slow diffusion), Theorem 1.3 (fast diffusion), and their global corollaries.","tokens_in":17732,"tokens_out":14767,"duration_ms":141512,"significance":"If the proof were complete, the paper would provide the first local and global gradient estimates of Li-Yau type for the general Leibenson equation on noncompact manifolds with negative Ricci lower bound, extending earlier porous-medium and p-Laplacian results. The paper is clearly written, compares its constants with known estimates, and treats the fast-diffusion case under a dimension condition p-nD>0. No fitted parameters appear, and the final estimates are explicit. However, the central Moser-iteration argument currently contains a sign/positivity gap, and several displayed constants are algebraically inconsistent, so the main result is not yet established as written.","major_comments":[{"comment":"The proof of (2.23) uses the inequality 2c1fφ ≥ 0, which is valid only when f≥0, and the subsequent iteration in Lemmas 2.4-2.8 forms powers f^{λ-1}, f^{λ/2}, f^λ, and f^{λ+1} for real λ, so the whole Moser iteration requires f to be nonnegative. No such lower bound is proved from the assumptions. The theorem's hypotheses do not supply one: for p=2, q>1, K=0, the pressure variable satisfies v_t = (q-1)v Δv + |∇v|², and the local solution with v(0,x)=1+|x|² has v=1, ∇v=0, and v_t=2n(q-1)>0 at x=0. Since φ≡0 for K=0, f=Fα=-α v_t/v<0 there, while (1.6) holds with suitable Λmin<1<Λmax. Thus the Caccioppoli inequality (2.26), the mean-value inequality (2.35), and Lemmas 2.8-2.9 do not apply to the actual f of the theorem. A standard repair would be to run the iteration on the positive part f_+ and prove the required differential inequality for f_+; as written this is a load-bearing gap.","section":"§2.1, Lemmas 2.2 and 2.4, and Theorem 2.9"},{"comment":"The KΛmax coefficient in the final estimates is computed incorrectly. From cδ=pδ/(p-1), c2=2(p-1)p(α-1)/(nδα²), and φ≤2b/a, the bound (2.25) should read φ(t) ≤ cδKΛmax/c2 = nδ²α²KΛmax/[2(p-1)²(α-1)], not α²nKδ 2Λmax/((p-1)(α-1)). The incorrect value is then propagated into (2.38), (2.39), (1.7), and (1.8), so the displayed explicit constants in the main theorems are not correct as printed. Additionally, the prefactor and exponent in (1.7) are inconsistent with those in (2.38): (1.7) places C0 in the exponent and omits the C1/c1 prefactor, while (2.38) has no C0 in the exponent and includes C1/c1 multiplicatively. The two statements are not equivalent as written.","section":"§2.1, Eq. (2.25), and Theorem 1.1 / Corollary 1.2"},{"comment":"The same positivity gap appears in the fast-diffusion part. In the proof of Lemma 3.3, after choosing the coefficients positive, the inequality is reduced to (3.47) by discarding the cross term involving fφ; this reduction is valid only if fφ≥0 (hence, since φ≥0, if f≥0). The subsequent estimates also use f^{λ}, f^{λ+1}, and f^{λ/2} for real λ. No pointwise lower bound for f is established from the assumptions (1.14), so the proof of Theorem 3.5 has the same unfilled hypothesis as the slow-diffusion argument.","section":"§3, Lemma 3.3 and Theorem 3.5"}],"minor_comments":[{"comment":"The word 'summond' should be 'summand'.","section":"§1, page 3"},{"comment":"The entry 'F ast diffusion case' contains a spacing typo and should read 'Fast diffusion case'.","section":"Table of contents"},{"comment":"The constant C0 in (1.8) is said to be as in Theorem 1.1, while Corollary 2.10 writes C1/c1; the two constants are equal in magnitude but the reader has to compare (2.39) and (1.8) to see this. It would be clearer to use one notation consistently.","section":"§1, page 2 and Corollary 2.10"},{"comment":"The notation B=B(x,r1) and B'=B(x,r2) for the two balls is slightly confusing because B was already used in the introduction for a geodesic ball of radius R; renaming the two balls B1 and B2 would improve readability.","section":"§2.3, Lemma 2.6"}],"recommendation":"major_revision","confidential_remarks":"The decisive issue is the sign/positivity gap in the Moser iteration; if the author can justify the iteration for the positive part of f and correct the constant inconsistencies, the paper is likely salvageable. I do not see a reason to reject on grounds of novelty or scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper makes a genuine advance: it proves Li-Yau type gradient estimates for the doubly nonlinear Leibenson equation ∂_t u = Δ_p u^q on noncompact manifolds with Ric ≥ -K, covering both slow diffusion (q(p-1)>1) and fast diffusion (q(p-1)<1). The fast diffusion general case appears to be new; previous results either handle p=2, closed manifolds, or nonnegative Ricci. The method — Moser iteration fed by a nonlinear Bochner formula from Wang-Chen and a mean value inequality — is technically solid, and the paper is well written.\n\nThe central soft spot is serious. In Lemma 2.2 the proof drops the term 2 c1 f φ using '2 c1 f φ ≥ 0'; Lemma 2.4 then works with f^λ for real λ and forms f^{λ+1}. This is only valid if f = Fα - φ is nonnegative, and that is never established. In fact it is false for admissible solutions: for the porous medium case p=2, q>1, take v(0,x)=1+|x|^2. At x=0, ∇v=0, v_t = 2n(q-1) > 0, so Fα = -α v_t/v < 0; since φ ≥ 0, f < 0. This is a smooth positive local solution satisfying the hypotheses. So the Caccioppoli inequality (2.26) and the mean value inequality (2.35) do not apply to the actual f. The gap looks repairable — replace f by f_+ in the test function, or observe that if f ≤ 0 the desired upper bound holds trivially — but the paper does not do that. This is a load-bearing issue, not a typo.\n\nThere are also minor inconsistencies in the KΛmax term: (1.8) and (2.25) have α^2 n K δ 2Λmax /((p-1)(α-1)), while (2.39) has α^2 n δ^2 K Λmax /((p-1)(α-1)) — a δ^2 and missing factor 2. Likely typos, but they should be fixed.\n\nThe restrictive assumption Λmin ≤ |∇v|^{p-2}v ≤ Λmax is used heavily (Λmin appears in denominators). The author honestly leaves open whether only an upper bound suffices. That is a limitation, not a flaw.\n\nThe citation pattern looks fine: the Bochner formula is cited from published work, and the Moser iteration lemma from the author's own [5] is a legitimate auxiliary tool, not self-citation in bad faith.\n\nBottom line: the result is probably true and worth pursuing, but as written the proof has a real gap. I would send it to referees — a knowledgeable referee can confirm the positive-part fix — but I would not trust the theorem from the arXiv version alone.","headline":"A genuine advance in Li-Yau estimates for the doubly nonlinear equation, but the Moser iteration as written has a sign gap that needs a positive-part repair before the theorem is proved.","tokens_in":18302,"tokens_out":5004,"would_cite":false,"duration_ms":39786,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K55","58J35","53C21","35B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves explicit Li–Yau type gradient estimates for positive solutions of the doubly nonlinear Leibenson equation on Riemannian manifolds with Ricci curvature bounded below by a negative constant.","keywords":["Leibenson equation","doubly nonlinear parabolic equation","p-Laplacian","gradient estimates","Li-Yau estimate","Moser iteration","Riemannian manifold","Ricci curvature"],"falsifier":"Compute, for the explicit Barenblatt self-similar solution of the porous medium equation ∂tu = Δu^q on R^n (so p = 2, Ric = 0), the quantity sup(|∇v|^2/v − α ∂t v/v) and compare it with the C0/t term in (1.8); if the supremum ever exceeds that bound, the global estimate is false, while matching it confirms the sharp 1/t decay in the known case.","tokens_in":17219,"feed_emoji":"📐","tokens_out":9562,"duration_ms":103938,"temperature":0.7,"pith_summary":"The paper proves Li–Yau type gradient estimates for positive solutions of the Leibenson (doubly nonlinear) equation ∂tu = Δp u^q on Riemannian manifolds whose Ricci curvature is bounded below by −K. The estimates control the quantity |∇v|^p/v − α ∂t v/v, where v is a power-like transform of u, by explicit expressions in time, the ball radius, the curvature, and the assumed bounds on |∇v|^{p−2}v. Both the slow diffusion case q(p−1) > 1 and the fast diffusion case q(p−1) < 1 are treated, with local estimates in geodesic balls and global estimates obtained by letting the radius tend to infinity. If the theorems are correct, this gives the first such estimates for the general doubly nonlinear equation on noncompact manifolds with a negative Ricci lower bound, extending earlier work that handled closed manifolds or the porous medium case p = 2.","feed_headline":"Gradient bounds for doubly nonlinear heat flow on manifolds","feed_subtitle":"Slow and fast diffusion both get explicit Li–Yau bounds under a negative Ricci lower bound.","key_machinery":"The argument is carried by a change of variables v = (q(p−1)/δ) $u^{{δ/(p−1)}}$ with δ = q(p−1) − 1 in the slow case, and v = (q(p−1)/D) $u^{{−D/(p−1)}}$ with D = 1 − q(p−1) in the fast case, which puts the equation into a form where an operator F = ∂t − δ/(p−1) v L (or with D) can be applied, with L a p-Laplacian-type elliptic operator. A nonlinear Bochner-type inequality, imported from reference [17], gives a differential inequality for F acting on Fα = |∇v|^p/v − α ∂t v/v. The proof then runs a Moser iteration scheme, defined here as a repeated application of an L^λ mean value inequality and a Caccioppoli-type estimate, using Sobolev and Faber–Krahn inequalities on geodesic balls and an auxiliary function φ(t) that absorbs the curvature term. The lower bound Λmin appears in denominators throughout this iteration, which is why the argument excludes points where ∇v = 0.","core_discovery":"On its own terms, the central claim is that a Li–Yau type gradient estimate holds for positive smooth solutions of ∂tu = Δp u^q on complete Riemannian manifolds with Ric ≥ −K, K ≥ 0, in both the slow-diffusion regime q(p−1) > 1 (Theorem 1.1) and the fast-diffusion regime 1 > q(p−1) with p − nD > 0 (Theorem 1.3). For the slow case, the estimate bounds sup_B (|∇v|^p/v − α ∂t v/v) by the explicit right-hand side (1.7), and the global analogue (1.8) is obtained by sending R → ∞. For the fast case, the analogous bound controls |∇v|^p/v + α ∂t v/v and yields the global estimate (1.16). The proofs require the two-sided pointwise bound Λmin ≤ |∇v|^{p−2}v ≤ Λmax in the relevant cylinder, and the author explicitly leaves open whether the lower bound can be relaxed to just the upper bound.","pith_inferences":["A natural testable extension is whether the two-sided bound can be weakened to the upper bound alone; the author leaves this open, and the proof's only obstruction appears to be the denominator Λmin in the Moser iteration.","Li–Yau estimates typically imply parabolic Harnack inequalities; if the same reasoning carries through here, it would give quantitative control of ratios of positive solutions of the doubly nonlinear equation, which the paper does not state explicitly.","For p = 2, the estimates should reduce to the known porous-medium results of references [12] and [9]; checking this limit is a consistency test for the constants and would clarify how the general p case improves on the p = 2 machinery.","Because all analytic steps are local, a plausible further direction is to run the same argument for doubly nonlinear equations with lower-order terms or on weighted manifolds, though that is not attempted in the paper."],"forward_implications":["On a geodesically complete manifold with Ric ≥ −K, every positive smooth solution satisfying the two-sided bound obeys the global estimate (1.8) in the slow-diffusion case, with the right-hand side depending explicitly on t, K, Λmin, Λmax, and α.","Sending R → ∞ removes the dependence on the ball radius and leaves the curvature entering through the term α^2 n K δ^2 Λmax / ((p−1)(α−1)), so the estimates are stable as K → 0.","In the fast-diffusion regime with p − nD > 0, the same machinery gives the global bound (1.16), which the paper notes is the first such estimate for the general doubly nonlinear equation on manifolds with Ric ≥ −K.","The constants are explicit in p, q, n, and α, and the geometry of the ball enters only through the Sobolev constant, so the estimates are quantitative rather than qualitative."],"supporting_citations":[{"why":"Supplies the nonlinear Bochner-type inequality for the operator F applied to Fα, the starting point for both the slow and fast diffusion proofs.","marker":"[17]"},{"why":"Provides the Moser iteration scheme for gradient estimates of porous medium equations, which the paper adapts to obtain its local estimates.","marker":"[9]"},{"why":"The original Li–Yau gradient estimate for the heat equation that this paper generalizes to the doubly nonlinear setting.","marker":"[11]"},{"why":"Earlier global estimates for the porous medium equation (p = 2), serving as the baseline that the general-p result extends and compares against.","marker":"[12]"},{"why":"The Moser iteration method and parabolic mean value inequality used in the iteration step of the proof.","marker":"[14]"},{"why":"Provides the cylinder Moser inequality and the auxiliary iterative lemma (Lemma 4.5) used in the proof.","marker":"[5]"},{"why":"The previous closed-manifold result for the Leibenson equation, which the global estimates here extend to noncompact manifolds.","marker":"[2]"},{"why":"The source of the local gradient estimate idea for p-harmonic functions that underlies the Moser argument used here.","marker":"[16]"}],"fun_headline_variants":["Li-Yau bounds for doubly nonlinear diffusion on manifolds","Gradient estimates for slow and fast nonlinear diffusion on manifolds","Doubly nonlinear heat flow: explicit gradient bounds with negative Ricci","Li-Yau type estimates for both slow and fast Leibenson diffusion","Gradient bounds for doubly nonlinear p-Laplacian heat flow"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof needs the uniform lower bound Λmin ≤ |∇v|^{p−2}v on the whole cylinder, because Λmin appears in denominators in the Moser iteration; at points where ∇v = 0 this condition fails, and the author leaves open whether the upper bound Λmax alone would suffice.","fun_headline_variants_meta":{"raw":{"variants":["Li-Yau bounds for doubly nonlinear diffusion on manifolds","Gradient estimates for slow and fast nonlinear diffusion on manifolds","Doubly nonlinear heat flow: explicit gradient bounds with negative Ricci","Li-Yau type estimates for both slow and fast Leibenson diffusion","Gradient bounds for doubly nonlinear p-Laplacian heat flow"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00069,"raw_usage":{"total_tokens":3078,"prompt_tokens":851,"completion_tokens":2227,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":467,"completion_tokens_details":{"reasoning_tokens":2141}},"tokens_in":467,"tokens_out":2227,"duration_ms":18984,"temperature":1.0,"reasoning_tokens":2141,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:40:19.614371+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for the explicit Barenblatt self-similar solution of the porous medium equation ∂tu = Δu^q on R^n (so p = 2, Ric = 0), the quantity sup(|∇v|^2/v − α ∂t v/v) and compare it with the C0/t term in (1.8); if the supremum ever exceeds that bound, the global estimate is false, while matching it confirms the sharp 1/t decay in the known case.","supporting_citations":[{"cited_title":"Wang and W","cited_arxiv_id":null,"evidence_quote":"Supplies the nonlinear Bochner-type inequality for the operator F applied to Fα, the starting point for both the slow and fast diffusion proofs."},{"cited_title":"Huang and B","cited_arxiv_id":null,"evidence_quote":"Provides the Moser iteration scheme for gradient estimates of porous medium equations, which the paper adapts to obtain its local estimates."},{"cited_title":"Li and S","cited_arxiv_id":null,"evidence_quote":"The original Li–Yau gradient estimate for the heat equation that this paper generalizes to the doubly nonlinear setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier global estimates for the porous medium equation (p = 2), serving as the baseline that the general-p result extends and compares against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The Moser iteration method and parabolic mean value inequality used in the iteration step of the proof."},{"cited_title":"Grigor’yan and P","cited_arxiv_id":null,"evidence_quote":"Provides the cylinder Moser inequality and the auxiliary iterative lemma (Lemma 4.5) used in the proof."},{"cited_title":"Chen and C","cited_arxiv_id":null,"evidence_quote":"The previous closed-manifold result for the Leibenson equation, which the global estimates here extend to noncompact manifolds."},{"cited_title":"Wang and L","cited_arxiv_id":null,"evidence_quote":"The source of the local gradient estimate idea for p-harmonic functions that underlies the Moser argument used here."}],"review_version":1}