{"id":"248bfa9f-e567-4489-9f9f-f3dc9c10f56f","arxiv_id":"2607.10384","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Constant states (A,A) of the fully parabolic Keller-Segel system are nonlinearly stable for A≤1 and unstable for A>1, with heat or half-heat decay rates.","lead":"The paper proves a sharp critical threshold A_crit=1 for nonlinear stability of constant equilibria in the fully parabolic Keller-Segel system on R^d. This closes the stability-instability dichotomy (including the critical case) and supplies matching heat-type decay rates.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly flags the unproved invariance under τ,γ as the softest sentence in the introduction, yet that sentence is not load-bearing for the theorems actually proved (which explicitly fix τ=γ=1). The spectral threshold, the Fourier-space smallness arguments, the energy estimates that close for A≤1, and the carefully tuned unstable-mode construction for A>1 are all self-contained and free of free parameters or circular reasoning. Consequently the ACCEPT verdict stands; the only adjustment is to treat the general-τ,γ claim as an informal remark rather than part of the main result.","tokens_in":39427,"tokens_out":418,"duration_ms":24229,"concrete_test":"Re-derive the characteristic polynomial of LA(ξ) for A=1 and A=1.1; confirm that minξ Re λ−(|ξ|) changes from 0 to a strictly negative value precisely at A=1, and that the escape-time lower bounds in (5.2) remain positive for the chosen constants C0,η0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central dichotomy (Theorems 1.1–1.3) is proved only for the normalized system with τ=γ=1, where the linear operator LA(ξ) has eigenvalues λ± whose sign is controlled exactly by A−1 (Section 2). All a-priori estimates (Lemmas 3.1–3.2, 4.1–4.3), global existence via smallness of Fourier L1 norms, decay rates, and the unstable-mode construction (Section 5) close rigorously under that normalization. The parenthetical claim that the same threshold holds for arbitrary τ,γ>0 is not needed for the stated theorems and is therefore peripheral; the analytic arguments themselves contain no hidden gaps, circular steps, or uncontrolled remainders.","agreement_with_reader":"partial"},"referee_report":{"model":"grok-4.5","summary":"The paper studies the Cauchy problem for the fully parabolic Keller–Segel system (1.1) with τ = γ = 1 near constant equilibria (A, A). After rewriting the system for the perturbations (n, c) = (b − A, a − A), the authors perform a Fourier-space spectral analysis of the linear operator L_A(ξ) and identify the critical value A_crit = 1 from the sign of the eigenvalue λ_−. They prove nonlinear Lyapunov stability for A < 1 (Theorem 1.1) and A = 1 (Theorem 1.2) under smallness assumptions on the Fourier L^1 (and L^∞) norms of the initial data, together with heat-type and half-heat-type decay rates for (n, c) and faster rates for n − c. For A > 1 they construct initial data concentrated near an unstable Fourier mode and prove nonlinear instability via an escape-time argument (Theorem 1.3). Local existence, a-priori estimates, and the Duhamel formulae are developed in Sections 2–5 and the appendices.","tokens_in":39605,"tokens_out":834,"duration_ms":6214,"significance":"The work supplies a complete nonlinear stability–instability dichotomy for the fully parabolic Keller–Segel system around constant states on R^d, including the critical case A = 1 that had remained open even for the parabolic–elliptic counterpart. The spectral threshold A_crit = 1 is parameter-free and arises cleanly from the linear symbol; the subsequent nonlinear estimates close rigorously. The paper also shows that the classical critical-mass phenomenon disappears when γ > 0 (Remark 1.3) and obtains sharp asymptotic rates that distinguish the subcritical and critical regimes. These results fill a genuine gap between the parabolic–elliptic theory of Cygan et al. and the fully parabolic setting, and the detailed Fourier-Lebesgue and Sobolev estimates are of independent technical interest.","major_comments":[],"minor_comments":[{"comment":"Page 2 (after (1.1)): the claim that τ, γ > 0 “do not play any role” and may be set to 1 without loss of generality is left unproved. While the stated theorems concern only the normalized system, a short remark or reference explaining the invariance of the threshold would remove any ambiguity.","section":null},{"comment":"Definition A (nonlinear Lyapunov stability): the pair of spaces ⟨X, Z⟩ is introduced but never specialized; it would help the reader if the concrete spaces used in Theorems 1.1–1.3 were identified explicitly with X and Z.","section":null},{"comment":"Lemma 2.1 / Remark 2.1: the refined blow-up criterion (2.1) is stated without proof; a one-line energy estimate for ∥c∥_{H^{s+1}} would make the argument self-contained.","section":null},{"comment":"Throughout Sections 3–4 the generic constant C is allowed to depend on A, but this dependence is not always recorded; a uniform convention (e.g., C = C(A, d, s)) would improve readability.","section":null},{"comment":"Appendix A.5 (construction of unstable data): the cut-off radius ε(θ) is chosen so that λ_− ≤ −θ(A−1)^2/(4A) on the support; a brief numerical illustration of how small ε must be for a typical A would make the construction more transparent.","section":null}],"recommendation":"accept","confidential_remarks":"The parenthetical claim that the same threshold holds for all τ, γ > 0 is peripheral and does not affect the correctness of Theorems 1.1–1.3. The analytic core is solid; I see no reason to delay publication for a minor invariance remark."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper finishes the constant-equilibrium story for the fully parabolic Keller-Segel system on R^d. The new pieces are nonlinear Lyapunov stability at the critical value A=1 (Theorem 1.2), the matching half-heat decay rates, and a clean escape-time instability proof for A>1. Together they give the full stable/unstable dichotomy that Cygan et al. left open for the parabolic-elliptic case and that Guo-Hwang only partially treated in the fully parabolic setting.\n\nThe technical work is careful and self-contained. Spectral analysis of the Fourier symbol L_A produces the threshold A_crit=1 from the sign of λ_-. Subcritical and critical global existence rest on smallness of Fourier-Lebesgue norms that close the nonlinear term; the critical case needs the extra L^∞ control and weighted estimates in Lemmas 4.2-4.3. Decay rates follow by standard energy ODIs once the solution is global and small. The instability construction (Section 5) is the usual unstable-mode + escape-time argument, written out with explicit constants. Appendices fill the Fourier estimates. Local existence and blow-up criteria are standard.\n\nThe only soft spot is the parenthetical claim that τ and γ play no role and can be set to 1 without loss. All proofs are done only for that normalization; the invariance is not shown. It is peripheral to the stated theorems, which are correct as written for τ=γ=1. No circularity, free parameters, or hidden gaps appear in the actual arguments. Citation pattern is appropriate.\n\nThis is for specialists who already work on chemotaxis or aggregation equations and want the fully parabolic constant-state picture closed. It does not reorganize the field, but it is a clean, usable extension. I would send it to referees; the math is solid enough to deserve the time.","headline":"Solid, complete dichotomy for fully parabolic KS around constants, including the missing critical nonlinear stability and half-heat rates.","tokens_in":40169,"tokens_out":491,"would_cite":true,"duration_ms":7437,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K45","35B35","35B40","92C17"],"pacs":[],"model":"grok-4.5","headline":"Constant equilibria of the fully parabolic Keller–Segel system are nonlinearly stable precisely when the background density is at most 1, and unstable above it.","keywords":["Keller-Segel system","fully parabolic","nonlinear stability","nonlinear instability","critical threshold","asymptotic decay rates","chemotaxis","Cauchy problem"],"falsifier":"Construct, for some A > 1 and arbitrarily small initial data in the Sobolev space of the theorem, a global solution whose L2 norms of the density and chemoattractant perturbations remain smaller than any fixed positive constant for all time; or, for A = 1, exhibit a family of small data whose solutions fail to decay at least as fast as t to the power minus one-half times the heat rate.","tokens_in":40341,"feed_emoji":"🧬","tokens_out":691,"duration_ms":8405,"temperature":0.7,"pith_summary":"The paper settles a stability dichotomy for constant steady states of the fully parabolic Keller–Segel chemotaxis system on the whole space. After shifting to perturbations around a constant density A, the authors prove that the zero state is nonlinearly Lyapunov stable for A less than or equal to 1 and nonlinearly unstable for A greater than 1. The threshold is sharp: subcritical solutions decay at the same rate as the heat equation, while critical solutions decay at half that rate; the difference between the two components decays even faster. The same analysis shows that the presence of a linear damping term in the chemoattractant equation destroys any critical-mass blow-up threshold that exists when that damping vanishes. A sympathetic reader cares because the result supplies the first complete stable-versus-unstable picture for the fully parabolic model, including the critical case that remained open even for the simpler parabolic-elliptic reduction.","feed_headline":"Keller–Segel constants are stable exactly up to density 1","feed_subtitle":"Above that threshold the fully parabolic model is nonlinearly unstable; critical decay is half the heat rate.","key_machinery":"Spectral decomposition of the Fourier-space linear operator L_A(ξ) into eigenvalues λ±(|ξ|), whose lower branch λ– is non-negative precisely when A ≤ 1 and becomes negative on a frequency band when A > 1. This spectral gap (or its absence) controls both the a-priori estimates that close global existence for small data and the growing mode used to prove nonlinear instability.","core_discovery":"There exists a critical threshold A_crit = 1 such that the constant equilibrium (A,A) of the fully parabolic Keller–Segel system (with unit diffusion and degradation rates) is nonlinearly Lyapunov stable for every A ≤ 1 and nonlinearly unstable for every A > 1. The associated decay rates are heat-like when A < 1 and half heat-like when A = 1.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Keller-Segel constants stable precisely for densities A ≤ 1","Fully parabolic Keller-Segel: nonlinear stability fails above A=1","Critical density 1 marks stability threshold for Keller-Segel equilibria","At A=1 Keller-Segel decay rates drop to half the heat equation","Steady states (A,A) stable iff A ≤ 1 in fully parabolic Keller-Segel"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The claim that the same critical value 1 works for every positive diffusion and degradation rates rests on the assertion that those rates play no role, yet all spectral and energy estimates are carried out only after both rates have been set to 1.","fun_headline_variants_meta":{"raw":{"variants":["Keller-Segel constants stable precisely for densities A ≤ 1","Fully parabolic Keller-Segel: nonlinear stability fails above A=1","Critical density 1 marks stability threshold for Keller-Segel equilibria","At A=1 Keller-Segel decay rates drop to half the heat equation","Steady states (A,A) stable iff A ≤ 1 in fully parabolic Keller-Segel"]},"model":"grok-4.5","effort":"low","cost_usd":0.005064,"raw_usage":{"total_tokens":1350,"prompt_tokens":663,"num_sources_used":0,"completion_tokens":104,"cost_in_usd_ticks":50640000,"prompt_tokens_details":{"text_tokens":663,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":583,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":663,"tokens_out":104,"duration_ms":5045,"temperature":1.0,"reasoning_tokens":583,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T12:07:34.577678+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Construct, for some A > 1 and arbitrarily small initial data in the Sobolev space of the theorem, a global solution whose L2 norms of the density and chemoattractant perturbations remain smaller than any fixed positive constant for all time; or, for A = 1, exhibit a family of small data whose solutions fail to decay at least as fast as t to the power minus one-half times the heat rate.","supporting_citations":[],"review_version":1}