{"id":"0a5bfb5f-98c4-4c62-b3a3-d943efd8338c","arxiv_id":"2607.04492","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Irregular double-phase parabolic equations with non-divergence data admit unique strong solutions that transfer integrability from data to the flux, achieve higher gradient integrability, and possess second-order space regularity for all r ≥ 0 in the admissible range.","lead":"This paper proves that solutions to irregular double-phase parabolic equations with variable growth and non-divergence forcing inherit integrability from the data, gain higher gradient integrability, and acquire second-order spatial regularity. It extends the authors' prior results to the full range of initial integrability exponents r ≥ 0 under mild gap and coefficient conditions.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the gap condition (1.9) and the high integrability of \nabla a,\nabla b,at,bt as the weakest (but standard and openly declared) hypotheses. A careful re-reading of the absorption arguments in §§3–5 confirms that every lower-order term generated by differentiating the variable exponents or the modulating coefficients is controlled precisely by those hypotheses; when they hold, the estimates close. No additional soft spot—e.g., an unjustified passage to the limit, a missing uniqueness argument, or an over-claimed range of r—appears. The paper therefore supports its strongest claim under the stated assumptions, and the ACCEPT verdict with high confidence remains appropriate.","tokens_in":49069,"tokens_out":509,"duration_ms":6401,"concrete_test":"Independently re-derive the key absorption inequality of Lemma 5.1 (the differential inequality that yields (5.31)) under the three regimes of §5, starting from the Green formula (3.14) and the pointwise lower bound of Lemma 4.3, without invoking any estimate that already assumes the final higher-integrability conclusion; if the same constants C0,C1 appear, the chain is self-contained.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims of Theorems 2.1–2.3 rest on the structural hypotheses (1.3), (1.9), (1.11)–(1.12) and the range restriction (2.4) on r. These enter every absorption step (Lemmas 3.2–3.5, 4.1–4.5 and the three cases of §5) exactly as the reader notes, but they are stated explicitly, match the standard gap/coefficient-regularity package of the double-phase literature, and are used consistently. The a-priori estimates for the regularized classical solutions, the passage to the limit via Vitali and modular Fatou, and the identification of the second-order terms are complete; no hidden circularity or unjustified absorption appears. The extension to the full range r\r≥0 (including the borderline σ=N+2) is a genuine technical improvement of the authors’ earlier work, not a re-packaging. Consequently the load-bearing assumptions are the usual ones of the field and do not undermine the stated theorems.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper establishes existence, uniqueness and global regularity for the irregular double-phase parabolic problem (1.1) with variable exponents p,q and non-divergence forcing f. Under the structural hypotheses (1.3), (1.9), (1.11)–(1.12) and the range restriction (2.4) on the integrability parameter r, Theorems 2.1–2.3 assert that a unique strong solution u∈V_r(Q_T) exists and inherits the modular integrability of the initial datum: F((·,t),\nabla u(·,t))|\nabla u(·,t)|^{r+2}∈L^1(Ω) a.e. t, together with the higher integrability |\nabla u|^{2(min{p,q}-1)+r+s}∈L^1(Q_T) for every s∈(0,4/(N+2)) and the second-order regularity F(z,\nabla u)|\nabla u|^{(r+2)/2}∈L^2(0,T;W^{1,2}(Ω)). The argument proceeds by regularizing the flux and data, deriving uniform a-priori estimates for classical solutions (energy, time derivative, second-order derivatives via the Green formula and pointwise Hessian inequalities), and passing to the limit by Vitali and modular Fatou arguments. The results extend the authors’ earlier work [1] to the full range r≥0, including the borderline σ=N+2.","tokens_in":49250,"tokens_out":922,"duration_ms":7925,"significance":"The work supplies a complete existence–uniqueness–regularity theory for a class of double-phase parabolic equations with non-divergence data that had previously been treated only for r≥max{2,p^+,q^+}. The global Calderón–Zygmund transfer of integrability, the self-improving higher integrability of the gradient, and the second-order space regularity are obtained under the standard gap and coefficient-regularity package of the double-phase literature. The technical core—uniform estimates independent of the regularization parameters, careful absorption of lower-order terms arising from variable exponents and modulating coefficients, and identification of the second-order limits—is solid and improves upon the authors’ previous single-phase and double-phase results. The contribution is therefore a genuine and useful advance within the regularity theory of nonstandard-growth parabolic equations.","major_comments":[],"minor_comments":[{"comment":"The definition of the constant K(N,σ,p,q) on p. 6 is written with a missing parenthesis; the expression should be clarified so that the two competing terms inside the max are unambiguous.","section":null},{"comment":"In several places (e.g., the statement of Theorem 2.2 and the estimate (2.5)) the higher-integrability exponent is written as 2(min{p(z),q(z)}-1)+r+s; it would help the reader if the authors consistently used the notation s(z)=min{p,q} already introduced on p. 4.","section":null},{"comment":"The dependence of the constants C,C' on the data is stated repeatedly but never collected in a single list; a short remark after Theorem 2.3 listing the precise structural quantities that enter the constants would improve readability.","section":null},{"comment":"A few typographical inconsistencies appear (e.g., “Calder´on-Zygmund” versus “Calderón–Zygmund”, occasional missing spaces after commas in multi-line displays). These are purely cosmetic.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a natural and carefully executed continuation of the authors’ own recent series [1,28–30]. The self-citation is heavy but legitimate: the earlier papers supply independent a-priori estimates that are used as black boxes. The technical novelty (full range r≥0, non-divergence data, irregular coefficients) is real and the proofs appear complete. I see no reason to request further external validation or to delay publication."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is a careful, self-contained extension of the authors' 2026 JGA paper. The real advance is that they now get the global Calderón-Zygmund transfer, higher integrability of the gradient, and second-order space regularity for the full range r ≥ 0 (including the previously excluded interval near zero), and they do it for non-divergence data with irregular modulating coefficients. Theorems 2.1–2.3 are the statements that matter; the rest of the paper is the machinery that produces them.\n\nWhat they do well is the a-priori work. They regularize, derive uniform energy, time-derivative and second-order estimates via the Green formula and pointwise Hessian inequalities (Lemmas 4.2–4.5), absorb the lower-order terms that come from differentiating the variable exponents and coefficients, then pass to the limit with Vitali and modular Fatou. The three cases in §5 (depending on the integrability of f) are handled cleanly, and the range restriction (2.4) on r is sharp relative to the interpolation they use. The proofs are long but complete; I did not find a circularity or an unjustified absorption step.\n\nThe soft spots are the usual ones for this literature and are stated up front: the gap condition (1.9) that forces |p-q| ≤ 2eta/(N+2), the lower bound s^- > 2(N+1)/(N+2), and the high integrability of \nabla a, \nabla b, a_t, b_t. Those hypotheses enter every absorption argument, so if the gap is larger the estimates fail. That is not a hidden flaw; it is the standard package. Heavy self-citation to their earlier single-phase and double-phase papers is present, but those results are used as black boxes for independent estimates, not as circular crutches.\n\nThis is for people who already work on nonstandard-growth parabolic equations and need the full-range initial-integrability statement. It is not a broad-interest paper, but the math is solid and the claims are supported. I would send it to referees without hesitation; a serious editor should not desk-reject it. Worth citing if you are in the same subfield and need the r \to 0 endpoint.","headline":"Solid technical extension of the authors' own double-phase work to the full range r ≥ 0 under non-divergence data; the estimates check out and the usual gap assumptions are stated cleanly.","tokens_in":49893,"tokens_out":601,"would_cite":true,"duration_ms":8143,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K65","35K67","35B65","35K55","35K99"],"pacs":[],"model":"grok-4.5","headline":"Irregular double-phase parabolic equations transfer integrability from data to the flux and gain second-order regularity for the full range of initial integrability r ≥ 0.","keywords":["double-phase parabolic equations","variable exponents","Calderón-Zygmund estimates","higher integrability","second-order regularity","non-divergence data","Musielak-Orlicz spaces"],"falsifier":"Construct (or exhibit a counter-example for) a pair of Lipschitz exponents whose difference exceeds the stated gap, or coefficients whose derivatives lie in a lower L^d space, and check whether the modular of order r of the flux remains bounded for a.e. time; if it does not, the central transfer claim fails.","tokens_in":49936,"feed_emoji":"∂","tokens_out":782,"duration_ms":8165,"temperature":0.7,"pith_summary":"This paper proves that solutions of irregular double-phase parabolic equations with variable exponents and non-divergence forcing inherit the integrability of the initial data and the right-hand side in the sense of Calderón-Zygmund theory, even when the modulating coefficients are merely weakly differentiable and the growth exponents p and q may differ. Under a controlled gap between p and q, high enough integrability of the derivatives of the coefficients, and an L^σ forcing term, a unique strong solution exists whose double-phase flux remains integrable of the same order r as the initial datum for almost every time, the gradient gains a fixed amount of higher integrability, and the flux itself belongs to a second-order Sobolev space. The result removes the previous restriction that r had to be larger than the maximum of the exponents and thereby covers the whole natural range r ≥ 0. A sympathetic reader cares because these global estimates close the regularity theory for a model that appears in nonlinear elasticity with heterogeneous hardening and supply the compactness needed for further analysis.","feed_headline":"Double-phase parabolic equations gain full-range gradient regularity","feed_subtitle":"Global Calderón-Zygmund transfer and second-order estimates hold for every initial integrability r ≥ 0","key_machinery":"A two-stage approximation: first a uniformly parabolic regularization of the flux and data that produces classical solutions, then uniform a-priori estimates obtained by testing with the divergence of the weighted flux, followed by absorption via interpolation inequalities that control the lower-order terms generated by the variable exponents and the derivatives of the modulating coefficients.","core_discovery":"Under the structural hypotheses on the exponents, coefficients and data, the Dirichlet problem admits a unique strong solution that transfers the modular integrability of order r from the initial datum and the forcing to the double-phase flux for a.e. time, improves the integrability of the gradient by any amount less than 4/(N+2), and places the weighted flux in L^{2}(0,T;W^{1,2}(Ω)).","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Full-range CZ estimates for irregular double-phase parabolic equations","Global integrability transfer holds for every r≥0 in double-phase flows","Strong solutions gain second-order space regularity in non-divergence setting","Higher gradient integrability for irregular double-phase evolution problems","CZ theory covers full initial modular range for double-phase parabolics"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"The exponents cannot differ by more than a small multiple of 1/(N+2) and the space-time derivatives of the modulating coefficients must be integrable to a high enough power that depends on that gap; if either restriction fails the absorption arguments that close the estimates break down.","fun_headline_variants_meta":{"raw":{"variants":["Full-range CZ estimates for irregular double-phase parabolic equations","Global integrability transfer holds for every r≥0 in double-phase flows","Strong solutions gain second-order space regularity in non-divergence setting","Higher gradient integrability for irregular double-phase evolution problems","CZ theory covers full initial modular range for double-phase parabolics"]},"model":"grok-4.5","effort":"low","cost_usd":0.007702,"raw_usage":{"total_tokens":2069,"prompt_tokens":1178,"num_sources_used":0,"completion_tokens":93,"cost_in_usd_ticks":77020000,"prompt_tokens_details":{"text_tokens":1178,"audio_tokens":0,"image_tokens":0,"cached_tokens":0},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":798,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":1178,"tokens_out":93,"duration_ms":8006,"temperature":1.0,"reasoning_tokens":798,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T18:38:59.626402+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Construct (or exhibit a counter-example for) a pair of Lipschitz exponents whose difference exceeds the stated gap, or coefficients whose derivatives lie in a lower L^d space, and check whether the modular of order r of the flux remains bounded for a.e. time; if it does not, the central transfer claim fails.","supporting_citations":[],"review_version":1}