{"id":"ec953088-929e-4d9c-a0a1-ae17c0528258","arxiv_id":"2607.11165","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"For large n, a codimension-n Lipschitz manifold of nonradial data produces finite-time blow-up to the homogeneous self-similar profile Φ_n for the inhomogeneous heat equation in R^3 with p>5.","lead":"This paper builds a finite-codimension family of initial data for a 3D nonlinear heat equation with a bounded spatial source so that solutions blow up in finite time with a prescribed self-similar profile. It shows the homogeneous blow-up stability mechanism survives even when exact scaling symmetry is broken.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The Reader correctly isolates the only external load-bearing input (the spectral theory of Φ_n for large n) and correctly judges that it does not undermine the paper’s own contribution. The non-autonomous perturbation is handled by a standard large-s_0 argument that is fully written out; the Lipschitz dependence of the manifold and of the blow-up time follows from the same estimates without additional assumptions. Consequently no adjustment of the ACCEPT verdict is warranted.","tokens_in":22390,"tokens_out":545,"duration_ms":5301,"concrete_test":"Verify that the source contribution in the energy identity (3.44) and the difference estimate (3.68) is absorbed by the spectral-gap term once s_0 ≫ 1 is chosen depending only on ∥f∥_∞, μ and c_n; if the resulting constant still permits the Gronwall bound (3.60) and the outgoing vector-field condition at s_*, the bootstrap closes independently of any further property of Φ_n.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1.1 is a closed bootstrap-plus-topological-selection argument that treats the spectral theory of the homogeneous profiles Φ_n (Collot–Raphaël–Szeftel, Props. 1.1 & 3.1) as a black-box input. That dependence is correctly flagged by the Reader, but it is not a soft spot internal to the present paper: the profiles and their spectral gap are already established for all sufficiently large n, and the paper never claims to re-prove them. The genuinely new analytic content—control of the non-autonomous forcing term λ^{2p/(p-1)}f(λ y+x(t)) by taking s_0 large enough so that it remains lower-order throughout the bootstrap (see the estimates after (3.32), (3.44), (3.47), (3.55) and the difference estimates (3.68)–(3.71))—is carried out carefully and does not rely on any hidden smallness of ∥f∥_∞. No circularity, missing estimate, or unjustified passage appears in the modulation, energy, L^∞ or Lipschitz arguments.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies the inhomogeneous nonlinear heat equation ∂_t u − Δu = |u|^{p−1}u + f(x) in R^3 for p > 5 and f ∈ L^∞ ∩ C^{0,1}(R^3). Building on the countable family of radial self-similar profiles Φ_n (and their spectral theory) constructed by Collot–Raphaël–Szeftel for the homogeneous equation, it constructs, for every sufficiently large n, every fixed x_0 and every sufficiently small λ_0 > 0, a codimension-n Lipschitz manifold M_{n,λ_0,x_0} ⊂ L^∞(R^3) of (generally non-radial) initial data whose solutions blow up in finite time T ≍ λ_0^{2}. The solutions admit a modulated self-similar decomposition whose remainder vanishes in L^∞ as t → T, the modulation center converges to a blow-up point, and the blow-up-time map is Lipschitz continuous with respect to the L^∞ topology on the manifold. The argument proceeds by geometric decomposition, modulation ODEs, weighted energy estimates exploiting the spectral gap, L^∞ comparison, bootstrap improvement, and a Brouwer fixed-point selection of the unstable coefficients; the non-autonomous forcing generated by f is controlled as a lower-order term by taking the initial renormalized time large.","tokens_in":22684,"tokens_out":951,"duration_ms":16313,"significance":"If correct, the result shows that the finite-codimensional stability mechanism for type-I self-similar blow-up developed for the homogeneous supercritical heat equation remains robust under a bounded Lipschitz spatial inhomogeneity that completely breaks scaling and translation invariance. No smallness of ∥f∥_∞ is required; the perturbation is rendered lower-order by a sufficiently small initial scale. This is a genuine advance beyond previous constructions that rely on exact scaling, and the framework is expected to apply to other parabolic problems lacking scaling symmetry. The paper correctly treats the existence and spectral properties of Φ_n as black-box input from Collot–Raphaël–Szeftel (2019) and supplies a complete, self-contained bootstrap-plus-topological argument for the inhomogeneous perturbation, including Lipschitz dependence of the manifold and of the blow-up time.","major_comments":[],"minor_comments":[{"comment":"In the statement of Theorem 1.1 and the definition of M_{n,λ_0,x_0} the authors note that higher regularity of the graphing functions a_j remains open; a brief remark on whether the present Lipschitz estimates can be bootstrapped to C^{1,α} (or why they cannot) would help the reader assess the sharpness of the result.","section":null},{"comment":"Section 3.1, Lemma 3.1: the map F is introduced as an L^∞-valued expression, yet the subsequent implicit-function argument is applied only to the finite-dimensional map G. A one-sentence clarification that G is smooth as a map into R^{n+4} (while F itself need not be viewed as a smooth Banach-space map) would remove a possible source of confusion.","section":null},{"comment":"Equation (3.18) and the subsequent estimates for NL: the Taylor remainder is written ≲ v^{2} + v^p; since p > 5 the quadratic term dominates for small v, but it would be cleaner to record the precise range of validity (e.g., |v| ≤ 1) once and for all.","section":null},{"comment":"References [14] and [7] appear as arXiv preprints with future dates; updating the bibliographic data (or marking them as preprints) would improve permanence.","section":null},{"comment":"Notation: the same symbol λ is used both for the modulation parameter and, earlier, for the scaling family of the homogeneous equation; a typographic distinction (e.g., λ(t) versus λ_scale) would avoid momentary ambiguity in Section 1.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a solid, carefully written contribution that sits comfortably within the scope of a strong analysis journal. The dependence on Collot–Raphaël–Szeftel is properly acknowledged and does not diminish the novelty of the inhomogeneous analysis. I see no reason to request major changes."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is that Zhang shows the finite-codimension self-similar blow-up manifolds of Collot–Raphaël–Szeftel survive a bounded Lipschitz source term, even though scaling is broken. That is the actual novelty: after dynamical rescaling the forcing becomes λ^{2p/(p-1)} f(λy + x(t)), and by taking the initial scale small enough (s0 large) it stays lower-order throughout the bootstrap with no smallness assumption on ||f||_∞.\n\nWhat the paper does well is the closed argument. Geometric decomposition (Lemma 3.1), modulation ODEs (3.3), weighted energy with the spectral gap (3.4), L^∞ comparison (3.5), improved bounds, and a Brouwer selection of the unstable coefficients are all written carefully and self-contained once the homogeneous profiles are granted. The difference estimates that give Lipschitz dependence of the manifold and of the blow-up time are also clean. The citation pattern is honest: the profiles Φ_n, eigenvalues, and gap constant c_n are imported as black-box input from CRS 2019 and never re-proved, which is the right way to do it.\n\nSoft spots are minor and proportional. Everything rests on the existence and spectral theory of Φ_n for large n; if those fail the whole construction collapses, but that is already established and the paper does not pretend otherwise. The manifold is only Lipschitz, not C^1, which the author flags as open. The result is restricted to d=3, p>5 and large n, so it is subfield-limited. None of these are load-bearing flaws inside the present argument.\n\nThis is for people who work on singularity formation for semilinear parabolic equations and care about robustness under broken scaling. Specialists will get value; outsiders will not. The math is solid, the estimates check out, and a serious editor should send it to referees. I would engage with it if I were working near this program.","headline":"Solid, carefully executed extension of Collot–Raphaël–Szeftel to a non-autonomous source; the new analytic content is real and the bootstrap closes.","tokens_in":23270,"tokens_out":518,"would_cite":true,"duration_ms":7271,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B44","35C06","35B35","35K57","35Q92"],"pacs":[],"model":"grok-4.5","headline":"A bounded Lipschitz source does not destroy finite-codimension self-similar blow-up for the nonlinear heat equation in three dimensions.","keywords":["finite-time blow-up","inhomogeneous nonlinear heat equation","self-similar profiles","finite-codimension stability","modulation","spectral gap","broken scaling invariance"],"falsifier":"If, for some large n used in the construction, the linearized operator around Φ_n fails to have the claimed simple unstable eigenvalues and spectral gap, or if numerical or analytic integration of the renormalized equation with a non-zero f shows that the remainder does not stay O(e^{-μs}) when λ_{0} is small, the bootstrap and topological selection collapse.","tokens_in":23289,"feed_emoji":"🔥","tokens_out":863,"duration_ms":9199,"temperature":0.7,"pith_summary":"This paper shows that adding a bounded, Lipschitz space-dependent forcing term to the supercritical nonlinear heat equation does not destroy the finite-codimension self-similar blow-up mechanism known for the homogeneous equation. For every sufficiently large integer n, every prescribed location, and every sufficiently small initial scale, there is a codimension-n Lipschitz manifold of nonradial initial data whose solutions blow up in finite time at a rate controlled by that scale. After rescaling, the profiles converge to a prescribed self-similar profile of the homogeneous problem, and the blow-up point and blow-up time depend Lipschitz-continuously on the data. The point is that exact scaling invariance, which most earlier constructions rely on, is broken by the source; the paper proves the non-autonomous perturbation remains lower order if the initial scale is small enough, so the same spectral-gap and topological-selection argument still closes. A sympathetic reader cares because many realistic models include inhomogeneous terms that break scaling, and this supplies a template for carrying precise singularity constructions into that larger class.","feed_headline":"Source terms do not kill self-similar heat blow-up","feed_subtitle":"A codimension-n manifold of data still blows up with the homogeneous profile when scaling is broken","key_machinery":"The finite-codimension stability mechanism of Collot–Raphaël–Szeftel, adapted to a non-autonomous renormalized flow: geometric decomposition around Φ_n, modulation of scale and center, spectral-gap energy estimates on the orthogonal remainder, L^∞ bounds via parabolic comparison, and topological selection of the unstable coefficients by a Brouwer fixed-point argument that keeps the trajectory inside a bootstrap tube.","core_discovery":"For the equation ∂_t u − Δu = |u|^{p−1}u + f(x) in R^{3} with p > 5 and f bounded and Lipschitz, and for every sufficiently large n, every fixed x_{0} and every sufficiently small λ_{0} > 0, there exists a codimension-n Lipschitz manifold of nonradial initial data such that the corresponding solutions blow up in finite time T ≃ λ_{0}^{2}, the rescaled profile converges in L^∞ to the prescribed homogeneous self-similar profile Φ_n, the modulation center converges to a blow-up point, and the blow-up-time map is Lipschitz.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Inhomogeneous sources preserve self-similar blow-up manifolds","Codim-n data still blow up under bounded Lipschitz sources","Self-similar heat blow-up robust to spatial inhomogeneity","Finite-codim stability holds despite broken scaling invariance","Homogeneous profiles survive for inhomogeneous parabolic blow-up"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The whole argument treats as given the existence, smoothness and spectral-gap properties of the countable family of radial self-similar profiles Φ_n and their eigenfunctions, which are known only for sufficiently large n.","fun_headline_variants_meta":{"raw":{"variants":["Inhomogeneous sources preserve self-similar blow-up manifolds","Codim-n data still blow up under bounded Lipschitz sources","Self-similar heat blow-up robust to spatial inhomogeneity","Finite-codim stability holds despite broken scaling invariance","Homogeneous profiles survive for inhomogeneous parabolic blow-up"]},"model":"grok-4.5","effort":"low","cost_usd":0.004622,"raw_usage":{"total_tokens":1391,"prompt_tokens":835,"num_sources_used":0,"completion_tokens":64,"cost_in_usd_ticks":46220000,"prompt_tokens_details":{"text_tokens":835,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":492,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":835,"tokens_out":64,"duration_ms":4334,"temperature":1.0,"reasoning_tokens":492,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T06:32:31.859715+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"If, for some large n used in the construction, the linearized operator around Φ_n fails to have the claimed simple unstable eigenvalues and spectral gap, or if numerical or analytic integration of the renormalized equation with a non-zero f shows that the remainder does not stay O(e^{-μs}) when λ_{0} is small, the bootstrap and topological selection collapse.","supporting_citations":[],"review_version":1}