{"id":"ee8daae7-55d4-43fe-81fd-d41cb03082f2","arxiv_id":"2506.14258","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A generalized parabolic De Giorgi class with unbalanced growth is shown to have locally bounded members under sharp integrability assumptions, with quantitative supremum estimates.","lead":"This paper proves that functions satisfying a broad family of nonlinear parabolic energy inequalities, including those coming from doubly nonlinear and nonstandard growth equations, are locally bounded without requiring extra integrability in the supercritical range. The result sharpens a classic line of De Giorgi and Ladyzhenskaya-Solonnikov-Uraltseva estimates and covers singular, degenerate, and limiting cases.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's proof is conditional on the standing assumption p<N; if p≥N is intended, the central claim is unproven.","rationale":"I checked the main components of the proof under the p<N assumption. The interpolation behind (3.7) is consistent: p* is chosen so that p* - αp = p(1+1/m)/N and the complementary exponent in the spatial Sobolev norm is αp, matching the time integral produced by Lemma 2.3. The level-gap estimates in (3.3) are justified using the fact that on {u ≥ k_{j+1}} the truncated powers are comparable up to constants 2^{jγ}. The L=M limiting argument only needs u^{m+L} ∈ L^1_loc, which follows from the membership in the class: for Λ>1, u^{m+Λ} ∈ L^{p_i}(L^{p_i}) for the maximizing i, hence in L^1 on bounded cylinders; for Λ≤1, u^{m+1} ∈ C(L^{1+m}). The abstract's phrase 'sub-critical ... no extra-integrability condition is needed' conflicts with Theorem 1.2, which requires u∈L^s for subcritical L>M; this appears to be a wording error (the super-critical case L<M is meant) and should be corrected, but it does not undermine the proof. The reader's conditional verdict already captures the p<N concern, so no change is needed.","tokens_in":16805,"tokens_out":35398,"duration_ms":343265,"concrete_test":"For the standard doubly nonlinear case m_i=m, p_i=q_i=p with p>N, check whether the anisotropic Sobolev embedding for p≥N (e.g., L^q-for-all-q or L^∞ type) can replace Lemma 2.3 and still close the recursion (3.9) with a finite exponent p* > 1+L/m. If no such embedding yields the same level-gap estimates, then Theorems 1.1 and 1.2 must be amended to include p<N as an explicit hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing restriction is the standing assumption p < N. Lemma 2.3, the only Sobolev embedding used in the proof, is stated for p < N. The exponent \\bar p = Nαp/(N-p) appearing in (3.7) is finite only in that range. The notation block writes 'p < N' once, but Theorems 1.1 and 1.2 do not list it as an explicit hypothesis, and the Section 4 examples (double-phase, Orlicz, anisotropic) do not state it. For p ≥ N, the chains in (3.7), (3.14), and (3.18) are not defined, so the proof does not cover that range. Since the abstract advertises the result for general doubly nonlinear, double-phase, Orlicz-type and fully anisotropic operators, the theorem's actual scope is narrower than claimed. This is a missing hypothesis rather than an internal contradiction; the iteration under p < N appears consistent.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a parabolic De Giorgi class PDG+(Ω_T, C) governed by an unbalanced energy inequality (1.9) with mixed (m_i, p_i) and (n_i, q_i) exponents. The main theorems assert local boundedness of non-negative functions in this class: Theorem 1.1 covers the super-critical and limiting regimes L := max(1,Λ) < M and L = M without extra integrability assumptions beyond membership in the class, while Theorem 1.2 covers L ≥ M under an additional L^s integrability condition with the sharp threshold (1.14). Quantitative sup estimates are stated in (1.12) and (1.15). Section 4 shows that weak subsolutions of doubly nonlinear, generalized Orlicz, and fully anisotropic parabolic equations satisfy (1.9). The proof is a De Giorgi iteration using an anisotropic Sobolev embedding (Lemma 2.3) and a geometric convergence lemma.","tokens_in":17001,"tokens_out":32890,"duration_ms":263407,"significance":"If the p<N restriction is made explicit and the quantitative estimates are corrected, the results are significant: they remove extra L^s integrability assumptions in the super-critical and limiting regimes for a broad class of singular and degenerate parabolic equations, including new sub-critical cases even for the standard p-Laplacian. The paper provides a unified class and verifies in Section 4 that concrete weak subsolutions belong to it, so the theorems are not circular. The De Giorgi iteration is standard, but the exponent bookkeeping is intricate and appears internally consistent under the p<N assumption. The explicit verification for Orlicz and anisotropic examples is a useful contribution.","major_comments":[{"comment":"The proof relies on Lemma 2.3, which is stated only for p<N, and the quantity \\bar p = Nαp/(N-p) appearing in (3.7) is finite only in that range. The notation block includes “p<N” once, but Theorems 1.1 and 1.2 do not list p<N as an explicit hypothesis, and the examples in Section 4 (especially the anisotropic case 4.3 with p_i possibly exceeding N) do not impose it. If the intended scope includes p≥N, the estimates (3.7), (3.14), and (3.18) are not defined as written. Please either state p<N explicitly in Theorems 1.1 and 1.2 and in the examples, or provide a separate embedding argument covering p≥N.","section":"Section 2.1, Lemma 2.3, Theorems 1.1–1.2"},{"comment":"The quantitative estimates as stated contain H(θ,ρ) to the first power inside the bracket, but the proof at (3.10) and (3.20)–(3.22) yields H^{(N+p)/p} inside the bracket, and the explicit formulas in Remark 3.1, equations (3.11)–(3.13), consistently use H^{(N+p)/p}. Thus (1.12) and (1.15) do not match the estimates actually proved. The statements should be corrected to [H(θ,ρ)^{(N+p)/p} ∫ u^...]^{...} to agree with the derivation and with the remark.","section":"Theorems 1.1 and 1.2, inequalities (1.12) and (1.15)"}],"minor_comments":[{"comment":"The display starts with “E_m^p = sup ...”, but the line is an inequality, not an equality; the label “E_m^p” is misleading and should be removed or replaced by a neutral label for the inequality.","section":"Definition 1.1, equation (1.9)"},{"comment":"The standing assumption p<N appears only in the line defining |λ/p|; since Lemma 2.3 and all of Section 3 depend on it, it should be listed as a hypothesis in Theorems 1.1 and 1.2 as well.","section":"Section 2.1"},{"comment":"The cutoff exponent γ0 is defined as q(1+1/(p_- α_-)), but the subsequent display uses ζ^{q/(p_- α_-)} inside the spatial integral while the first factor uses ζ^{γ0}_j; the relation between these cutoff powers should be clarified.","section":"Section 3.1.1, equation (3.7)"},{"comment":"The notation m and m_- is easy to confuse: in (4.4) the class parameter in (1.9) is m_-, while the exponent m in the coefficient u^{(m-m_-)(p-1)} plays the role of m_i. State this identification explicitly when translating conditions into (1.11).","section":"Section 4.2"}],"recommendation":"major_revision","confidential_remarks":"The exponent mismatch in (1.12) and (1.15) is likely a typo, since the proof and Remark 3.1 consistently use H^{(N+p)/p}; it should be corrected before publication. The more substantive issue is the unstated p<N restriction, which limits the advertised generality; the authors should either restrict the theorems and examples or extend the embedding argument. The core iteration appears sound under p<N, and the examples are genuinely verified, so I do not see grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper is a serious and useful piece of work. It defines a parabolic De Giorgi class (Definition 1.1) with unbalanced energy estimates covering doubly nonlinear, double-phase, Orlicz-type, and fully anisotropic equations, and proves local boundedness under sharp integrability assumptions. The key novelty is Theorem 1.1: in the supercritical parameter range L<M, boundedness follows from the energy inequality alone, without the extra L^s integrability that prior treatments of the p-Laplace and doubly nonlinear cases required. That is a genuine advance, and the proof is a careful De Giorgi iteration. I checked the exponent bookkeeping and the use of the anisotropic Sobolev embedding; it is consistent. Section 4 independently verifies that concrete subsolutions satisfy the class inequality, so there is no circularity.\n\nThe soft spots are real but manageable. First, the p<N restriction is stated once in the notation block and never repeated in Theorems 1.1-1.2. Lemma 2.3, the only embedding used, is stated for p<N; if the authors intend to claim results for p≥N, the argument does not cover it. That is more a presentation failure than a mathematical gap, but the abstract advertises the results broadly, so it should be flagged up front. Second, the abstract says 'no extra-integrability condition is needed', which is too generous. In the subcritical and critical range L≥M, Theorem 1.2 still requires u∈L^s_loc with s satisfying (1.14). What is removed there is the qualitative local boundedness assumption, not the L^s integrability. That distinction should be made explicit. Minor: Remark 1.2 assumes a chain rule tacitly; that is acceptable for the intended class but worth a sentence of justification.\n\nThe citation pattern is fine; self-citations are to earlier work on anisotropic/porous-medium equations and are used for background comparisons, not to hide anything.\n\nThis paper deserves a serious referee. The central claim holds up and the class definition is a useful contribution. I would recommend engaging with it, but the authors should be asked to state p<N in the theorems, temper the abstract, and clarify the exact role of the extra integrability in Theorem 1.2.","headline":"Genuine progress on sharp parabolic local boundedness; proof holds together, but the p<N hypothesis is under-advertised and the abstract overclaims the subcritical case.","tokens_in":17549,"tokens_out":3222,"would_cite":true,"duration_ms":30431,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B65","35B45","35K65"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that one unbalanced energy inequality is enough to force local boundedness of parabolic De Giorgi functions, with quantitative supremum bounds, and that extra $L^s$ integrability is only required in the subcritical range.","keywords":["parabolic De Giorgi classes","doubly nonlinear parabolic equations","local boundedness","nonstandard growth","anisotropic Sobolev embedding","unbalanced energy estimates","quantitative a priori estimates"],"falsifier":"One decisive check is to test the borderline case $L=M$ for the classical doubly nonlinear equation with $m=1$: Theorem 1.1 says every finite-energy subsolution in the class is locally bounded, as long as the $p<N$ embedding applies. If an explicit unbounded function can be exhibited that satisfies (1.9) and has finite $\\mathcal{V}_{\\mathrm{loc}}$ energy, the theorem is false; if, as the paper asserts, the known borderline blow-up examples fail the energy-class membership, the threshold is exactly sharp.","tokens_in":16612,"feed_emoji":"📐","tokens_out":15260,"duration_ms":146596,"temperature":0.7,"pith_summary":"This paper introduces a parabolic De Giorgi class $\\mathcal{PDG}^+$ whose defining feature is a single unbalanced energy inequality, and proves that every member of the class is locally bounded under an explicit condition relating the growth exponents. The condition is $L:=\\max(1,\\Lambda)\\le M$, where $\\Lambda$ measures the largest elliptic growth and $M=p|\\lambda/p|+(m+1)p/N$ is the threshold at which the parabolic and elliptic terms balance. When the inequality is strict, the proof yields a quantitative bound on the essential supremum; in the limiting case it still yields local boundedness. The interest is that the class contains local weak subsolutions to doubly nonlinear, double-phase/Orlicz-type, and fully anisotropic parabolic operators, so the result removes extra $L^s$ integrability assumptions that earlier work needed, even for the classical $p$-Laplacian in the subcritical cases.","feed_headline":"No extra integrability: De Giorgi class functions are locally bounded","feed_subtitle":"A single unbalanced energy inequality yields quantitative sup bounds for these parabolic equations.","key_machinery":"Three tools carry the argument. The two-sided bounds on the truncation $g(u^m,k^m)$ of Lemma 2.1 turn the energy inequality (1.9) into a clean recursion on level sets, with time term of exponent $1+1/m$ and spatial terms of exponents $p_i$ and $q_i$. The anisotropic Sobolev embedding of Lemma 2.3 then raises the integrability exponent on each superlevel set to $p_*=\\alpha p+(1+1/m)p/N$, where $\\alpha=(1/p)(1+(p/m)|\\lambda/p|)$; this is the step that produces the critical threshold $M$. Finally, the geometric convergence lemma makes the level-set integrals $y_j$ tend to zero once the initial integral is small enough. The strict supercritical case $L<M$ is exactly $p_*>1+L/m$, which makes the recursion super-linear; the limiting case $L=M$ is handled by choosing the starting level through absolute continuity of the integral; and the subcritical case recovers super-linearity from an $L^s$ assumption with $\\kappa_s>0$. The paper also defines the auxiliary quantities $H(\\theta,\\vec\\rho)$ and $R(\\theta,\\vec\\rho)$ that enter all quantitative bounds.","core_discovery":"The paper's central claim is that local boundedness is a property of the unbalanced energy estimate (1.9) itself, not of any particular equation or of extra integrability of the solution. More precisely, Theorem 1.1 states that if $u\\in\\mathcal{PDG}^+(\\Omega_T)$ and $L=\\max(1,\\Lambda)\\le M=p|\\lambda/p|+(m+1)p/N$, then $u$ is locally bounded; in the strict case the sup is controlled by an $\\int u^{m+L}$ average via (1.12). Theorem 1.2 covers the complementary subcritical range $L\\ge M$: assuming $u\\in L^s_{\\mathrm{loc}}$ for some $s$ with $\\kappa_s>0$, local boundedness follows with the quantitative estimate (1.15), where $\\kappa_s$ is the explicit non-degeneracy number (1.13). The paper also verifies that local weak subsolutions of the doubly nonlinear equation (1.3), the generalized Orlicz/double-phase equation (1.4), and the fully anisotropic equation (1.5) satisfy (1.9), so all these operators fall under the same theorem. The previously known unbounded borderline examples are noted to fail membership in the relevant Sobolev spaces, so they do not contradict the result.","pith_inferences":["A natural next step, not taken in the paper, would be to replace the $L^s$ hypothesis in Theorem 1.2 by a weaker Lorentz-space assumption; the proof's H\\\"older and Chebyshev steps suggest the optimal space is governed by the same exponent $\\kappa_s$, but this is an extrapolation.","Because $\\mathcal{PDG}^+$ is defined by an inequality rather than by an equation, I would expect the same bounds to hold for parabolic quasi-minimizers of the corresponding energies; the paper stops at weak subsolutions.","A testable extension would be to compute the sup bound from (1.12) for explicit self-similar profiles of the doubly nonlinear equation to see how sharp the constant factor is; that check is my suggestion, not something the paper reports."],"forward_implications":["For the standard doubly nonlinear $p$-Laplacian-type equation, local boundedness of non-negative weak subsolutions follows from the energy class alone whenever $m(p-1)+(m+1)p/N\\ge 1$, eliminating the extra $L^s_{\\mathrm{loc}}$ hypothesis used in previous borderline results.","For double-phase and Orlicz operators with growth satisfying (4.1), subsolutions are locally bounded under $\\max(1,n(q-1))\\le m(p-1)+(m_-+1)p/N$, with a quantitative sup bound in the strict case.","For fully anisotropic doubly nonlinear equations, the same conclusion holds under $\\max(1,\\Lambda)\\le p|\\lambda/p|+(m+1)p/N$, and Theorem 1.2 shows that in the subcritical range the usual qualitative local-boundedness assumption can be replaced by an $L^s$ assumption with sharp $\\kappa_s>0$.","The a priori estimates (1.12) and (1.15) provide explicit control of the essential supremum on interior cylinders in terms of the data, so they can be used as the first step toward continuity, Harnack inequalities, and further regularity for these classes."],"supporting_citations":[{"why":"Supplies the parabolic energy-class framework and the iteration scheme the paper adapts to unbalanced growth.","marker":"[29]"},{"why":"Introduces the level-set energy-inequality method for local boundedness that underpins the proof.","marker":"[13]"},{"why":"States the anisotropic Sobolev embedding used as the key integrability-gain step in Lemma 2.3.","marker":"[36]"},{"why":"Provides the parabolic anisotropic Sobolev embedding version cited alongside [36].","marker":"[18]"},{"why":"Gives the doubly nonlinear energy estimates and the two-sided bounds for the truncation function $g$ on which the class definition rests.","marker":"[5]"},{"why":"Is the elliptic $(p,q)$-growth boundedness result under extra integrability that the paper's supercritical theorem refines and extends.","marker":"[32]"},{"why":"Contains the geometric convergence lemma and the standard De Giorgi machinery invoked throughout the proof.","marker":"[15]"}],"fun_headline_variants":["Local boundedness without extra integrability in De Giorgi classes","Unbalanced energy implies local boundedness for parabolic equations","Doubly nonlinear, double-phase, anisotropic: one theorem bounds all","Exact integrability assumptions suffice for local boundedness","Quantitative sup bounds without extra integrability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the anisotropic Sobolev embedding used at the decisive step is valid, and it is stated in the paper only for $p<N$; the main theorems do not list this restriction explicitly, so the conclusion is conditional on that range.","fun_headline_variants_meta":{"raw":{"variants":["Local boundedness without extra integrability in De Giorgi classes","Unbalanced energy implies local boundedness for parabolic equations","Doubly nonlinear, double-phase, anisotropic: one theorem bounds all","Exact integrability assumptions suffice for local boundedness","Quantitative sup bounds without extra integrability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001118,"raw_usage":{"total_tokens":4649,"prompt_tokens":934,"completion_tokens":3715,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":550,"completion_tokens_details":{"reasoning_tokens":3636}},"tokens_in":550,"tokens_out":3715,"duration_ms":27447,"temperature":1.0,"reasoning_tokens":3636,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:19:00.056088+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One decisive check is to test the borderline case $L=M$ for the classical doubly nonlinear equation with $m=1$: Theorem 1.1 says every finite-energy subsolution in the class is locally bounded, as long as the $p<N$ embedding applies. If an explicit unbounded function can be exhibited that satisfies (1.9) and has finite $\\mathcal{V}_{\\mathrm{loc}}$ energy, the theorem is false; if, as the paper asserts, the known borderline blow-up examples fail the energy-class membership, the threshold is exactly sharp.","supporting_citations":[{"cited_title":"Ladyzenskaya, N.A","cited_arxiv_id":null,"evidence_quote":"Supplies the parabolic energy-class framework and the iteration scheme the paper adapts to unbalanced growth."},{"cited_title":"De Giorgi, Sulla differenziabilit´ a e l’analiticit´ a delle estremali degli integrali multipli regolari, Mem","cited_arxiv_id":null,"evidence_quote":"Introduces the level-set energy-inequality method for local boundedness that underpins the proof."},{"cited_title":"D¨ uzg¨ un, S.J.N","cited_arxiv_id":null,"evidence_quote":"Provides the parabolic anisotropic Sobolev embedding version cited alongside [36]."},{"cited_title":"Ok, Regularity for double phase problems under additional integrability assumptions, Nonlinear Anal.194(2020) 111408","cited_arxiv_id":null,"evidence_quote":"Is the elliptic $(p,q)$-growth boundedness result under extra integrability that the paper's supercritical theorem refines and extends."},{"cited_title":"DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993","cited_arxiv_id":null,"evidence_quote":"Contains the geometric convergence lemma and the standard De Giorgi machinery invoked throughout the proof."}],"review_version":1}