{"id":"6bcea0e3-7462-4ef8-88ff-c6e1b73e8632","arxiv_id":"2607.01857","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Establishes refined L^p-based blow-up criteria for triangular SKT cross-diffusion systems via hierarchical structure and tame Sobolev estimates, and proves global existence of non-negative strong solutions for two-species logistic systems in d ≤ 2.","lead":"The paper develops a self-contained well-posedness theory for triangular cross-diffusion systems of Shigesada-Kawasaki-Teramoto type in C0([0,T]; Hs(Td)) using regularity estimates for scalar Kolmogorov equations, yielding refined blow-up criteria. This enables proofs of global existence for two-species logistic systems in dimensions d ≤ 2.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's assessment is limited to the abstract; the central claim's load-bearing steps (hierarchical decoupling and tame estimates) cannot be stress-tested for hidden assumptions without the detailed proofs. No manufactured concern is warranted.","tokens_in":1698,"tokens_out":226,"duration_ms":25250,"concrete_test":"Re-derive the L^p blow-up threshold from the tame estimates in §3 (or equivalent section) for the two-species logistic case and verify that the exponent p remains finite and independent of the Sobolev index s when d=2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract presents the triangular hierarchy as enabling both the L^∞ blow-up criterion and its refinement to a finite-p L^p criterion via tame composition estimates in Sobolev spaces. No internal inconsistency, unjustified assumption, or gap in the stated logic is detectable from the given material; the hierarchical dependence is explicitly invoked to justify sequential regularity estimates, and the polynomial-growth hypothesis is stated as the condition for the L^p refinement.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript develops a well-posedness theory in C^0([0,T];H^s(T^d)) for triangular cross-diffusion systems of Shigesada-Kawasaki-Teramoto type on the torus, based on regularity estimates for scalar Kolmogorov equations. The triangular (hierarchical) structure of the diffusion coefficients is used to obtain refined blow-up criteria: finite-time singularities can occur only through divergence of the L^∞ norm, and under polynomial growth of the nonlinearities this is further weakened to an L^p blow-up condition for some finite p via tame composition estimates in Sobolev spaces. As an application, global existence of non-negative strong solutions is proved for two-species systems with logistic reaction terms when d≤2.","tokens_in":1752,"tokens_out":457,"duration_ms":17527,"significance":"If the central claims hold, the work supplies measurably weaker obstructions to global existence than classical Sobolev-type criteria, which is of direct interest for the long-time analysis of cross-diffusion models arising in mathematical biology. The explicit use of the triangular hierarchy to obtain sequential a-priori estimates, together with the tame-estimate approach, constitutes a technical contribution that could extend to other structured systems.","major_comments":[],"minor_comments":[{"comment":"The abstract and introduction should state the precise range of s (e.g., s > d/2 + k) required for the Sobolev embedding and tame estimates to close; this is load-bearing for the well-posedness statement but appears only implicitly.","section":"Introduction / §2"},{"comment":"Clarify whether the tame estimates invoked in the proof of the L^p criterion are taken from a cited reference or derived in an appendix; if the latter, the derivation should be referenced by equation number in the main text.","section":"§3 / §4"},{"comment":"In the application section, verify that the logistic reaction terms satisfy the stated polynomial-growth hypothesis uniformly in the parameters; a short remark on the admissible range of the growth exponents would strengthen the statement.","section":"§5"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report.","responses":[],"tokens_in":1249,"tokens_out":47,"duration_ms":12988,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core advance is the Lp blow-up criterion that exploits the triangular hierarchy to get sequential estimates, then applies tame composition bounds in Sobolev spaces to weaken the obstruction from L∞ or full Sobolev norms. This is presented as new relative to earlier criteria, and the abstract logic holds together without obvious circularity.\n\nThe well-posedness theory in C0([0,T]; Hs) is self-contained and the application to global non-negative solutions for logistic reactions in low dimensions follows directly once the criterion is available. That part is concrete and useful inside the niche of SKT-type models.\n\nThe main limitation is that everything still requires polynomial growth of the nonlinearities, and the global-existence result stays restricted to d ≤ 2, which is already the regime where many of these systems are known to work. It is not obvious how much practical gain the Lp version delivers for typical biological parameters, and the full verification of the tame estimates would need checking in the proofs.\n\nThis is for readers already working on cross-diffusion systems or parabolic regularity in population models. It is narrow but the claims are specific enough to be worth referee time; the estimates look checkable and the triangular structure is used in a consistent way.","headline":"The paper refines the blow-up criterion to an Lp norm for triangular cross-diffusion systems via tame estimates and Kolmogorov regularity, giving a modest technical improvement that yields global existence for the two-species logistic case in d ≤ 2.","tokens_in":2235,"tokens_out":337,"would_cite":false,"duration_ms":15753,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Finite-time blow-up in triangular cross-diffusion systems requires the L infinity norm of the solution to diverge.","keywords":["triangular cross-diffusion","blow-up criteria","global existence","Shigesada-Kawasaki-Teramoto","Sobolev well-posedness","Kolmogorov equations","logistic reactions"],"falsifier":"Constructing a solution that blows up in finite time while its L∞ norm remains bounded would falsify the refined blow-up criterion.","tokens_in":2585,"feed_emoji":"","tokens_out":562,"duration_ms":35573,"temperature":0.7,"pith_summary":"The paper develops a well-posedness theory for triangular cross-diffusion systems of Shigesada-Kawasaki-Teramoto type based on estimates for scalar Kolmogorov equations. It establishes that singularities can only occur if the supremum norm diverges. For nonlinearities with polynomial growth, the blow-up condition is refined to divergence in some Lp norm, providing a weaker condition than classical Sobolev criteria. This hierarchical dependence allows proving global existence for two-species systems with logistic reactions in dimensions up to two.","feed_headline":"Blow-up only if supremum norm diverges in triangular diffusion systems","feed_subtitle":"Refined Lp criterion gives weaker condition than Sobolev, proving global solutions for two-species logistic models in d at most 2.","key_machinery":"The triangular hierarchical structure where each diffusion coefficient depends only on species of lower index, combined with regularity estimates for scalar Kolmogorov equations.","core_discovery":"Finite-time singularities can occur only through the divergence of the L∞(Td) norm of the solution. Assuming polynomial growth of the nonlinearities, this criterion is refined to an Lp-based blow-up condition for some finite exponent p, yielding a substantially weaker obstruction to global existence than classical Sobolev blow-up criteria. The proof uses refined tame estimates for composition in Sobolev spaces, and the theory applies to prove global existence of non-negative strong solutions for two-species systems with logistic-type reaction terms in dimensions d ≤ 2.","pith_inferences":["The hierarchical structure might enable similar Lp refinements in other multi-species models with triangular diffusion.","Numerical simulations could use the Lp criterion to detect blow-ups more efficiently than higher norm monitoring.","The global existence result for d ≤ 2 suggests the method could be adapted for systems with additional species under suitable growth conditions."],"forward_implications":["Global existence holds for non-negative strong solutions in two-species logistic systems when d ≤ 2.","Well-posedness is obtained in the space C^0([0, T]; H^s(T^d)).","Refined tame estimates for composition operators in Sobolev spaces support the blow-up criteria.","The Lp blow-up condition is substantially weaker than Sobolev-based ones under polynomial growth."],"fun_headline_variants":["Blow-up only through L∞ divergence in triangular diffusion","Lp criterion refines Sobolev blow-up for cross-diffusion","Global existence for logistic triangular systems in d≤2","Hierarchical structure yields Lp blow-up condition","Tame estimates enable Lp blow-up refinement in d≤2"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The diffusion coefficient of each species depends only on species of lower index, creating a hierarchical structure.","fun_headline_variants_meta":{"raw":{"variants":["Blow-up only through L∞ divergence in triangular diffusion","Lp criterion refines Sobolev blow-up for cross-diffusion","Global existence for logistic triangular systems in d≤2","Hierarchical structure yields Lp blow-up condition","Tame estimates enable Lp blow-up refinement in d≤2"]},"model":"grok-4.3","cost_usd":0.003996,"raw_usage":{"total_tokens":2035,"prompt_tokens":658,"num_sources_used":0,"completion_tokens":76,"cost_in_usd_ticks":39962000,"prompt_tokens_details":{"text_tokens":658,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1301,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":658,"tokens_out":76,"duration_ms":11568,"temperature":1.0,"reasoning_tokens":1301,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-03T09:53:34.119822+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Constructing a solution that blows up in finite time while its L∞ norm remains bounded would falsify the refined blow-up criterion.","supporting_citations":[],"review_version":1}