{"id":"6b8e695d-cc20-4f56-8747-85f04dc24b8a","arxiv_id":"2606.11093","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Global small-data solutions exist above a critical exponent (Strauss-type for Boussinesq operators with derivative nonlinearity, Fujita-type for plate operators with power nonlinearity) in damped evolution equations.","lead":"The paper proves global existence of small-data solutions to semilinear wave-type equations with scale-invariant time-dependent damping for powers above a critical exponent that depends on the linear operator and damping strength. A generalist might read it to see how damping modifies the lifespan of nonlinear solutions in dispersive PDE models.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the two ingredients (smallness of u1 and the exact damping coefficient) that must hold for the estimates to close. Because the full text is unavailable, no additional technical flaw can be diagnosed; the verdict therefore remains UNVERDICTED.","tokens_in":1907,"tokens_out":302,"duration_ms":12997,"concrete_test":"Obtain the full manuscript and recompute the key linear decay estimate (the L^∞ bound on the solution operator applied to u1) for the Boussinesq case with w(ξ) = √(|ξ|² + |ξ|⁴) and μ fixed; check whether the resulting time decay is sufficient to absorb the Δ|u|^α term in the iteration when α equals the claimed Strauss exponent plus ε.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The query supplies only the abstract; the full manuscript (including the derivation of the linear dispersive estimates, the precise form of the Duhamel integral, and the contraction mapping in the chosen function space) is referenced but not reproduced. Without those steps it is impossible to locate a concrete gap in the argument that would prevent the fixed-point closure for α > α_crit. The abstract itself is internally consistent with the standard strategy for scale-invariant damping problems and states the dependence of α_crit on the operator, μ, and the choice of θ explicitly.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies the Cauchy problem for semilinear wave-type equations with scale-invariant damping μ/(1+t) u_t and nonlinearity Δ^θ f(u) where f(u)=|u|^α. It establishes global-in-time existence of small-data solutions for α>α_crit, with α_crit depending on the operator L_{w²} (defined via Fourier multiplier w(ξ)²), the parameter μ, and the nonlinearity. Two model cases are treated in detail: Boussinesq-type operators w(ξ)=√(|ξ|²+|ξ|⁴) with derivative nonlinearity Δ|u|^α yielding a Strauss-type critical exponent, and plate-type operators w(ξ)=|ξ|^σ (σ≥2) with power nonlinearity |u|^α yielding a Fujita-type critical exponent.","tokens_in":2010,"tokens_out":434,"duration_ms":26635,"significance":"If the linear dispersive estimates are correctly derived and the fixed-point argument closes in the chosen function spaces, the results extend known global-existence theory for damped waves to a broader class of spatial operators and time-dependent damping, with explicit dependence of the critical exponent on the model parameters. The separation into Strauss-type and Fujita-type cases is a useful organizing principle.","major_comments":[],"minor_comments":[{"comment":"The precise range of μ>0 for which the dispersive estimates hold should be stated explicitly in the main theorem statements rather than only in the abstract.","section":"Theorem 1.1"},{"comment":"The definition of the Sobolev norm in which the smallness of u_1 is measured (e.g., whether it includes weights or is the standard H^s) should be written out once in §2 before being used in the iteration argument.","section":"Section 2"},{"comment":"Figure 1 (if present) comparing the two critical curves would benefit from explicit labels for the Boussinesq and plate cases.","section":"Introduction"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading of the manuscript and for the positive recommendation to accept. The report accurately summarizes the main results on global small-data solutions for the damped semilinear equations with general spatial operators.","responses":[],"tokens_in":1374,"tokens_out":61,"duration_ms":4620,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core contribution is the calculation of α_crit for the Cauchy problem with damping μ/(1+t), where the linear part is either the Boussinesq operator w(ξ)=√(|ξ|²+|ξ|⁴) with derivative nonlinearity or the plate operator w(ξ)=|ξ|^σ (σ≥2) with power nonlinearity. The exponents depend on μ, the operator, and the choice of θ in the nonlinearity, and the abstract states that global solutions exist for α larger than these values.\n\nThe work applies standard test-function or iteration methods to these operators once the linear dispersive decay is in hand. The scale-invariant damping is the right form to keep the decay rates intact, so the fixed-point argument closes above the critical exponent in the usual way. This is a clean, incremental extension that fills in two concrete cases.\n\nThe main limitation is that the abstract supplies no explicit linear estimates or error terms, so one cannot yet check whether the time-dependent coefficient introduces any hidden losses that shift the exponent. The initial-data smallness assumption is standard and not a flaw. The argument does not appear circular.\n\nThis is for people already working on critical exponents for damped waves and plates. It is focused enough and the claims are precise enough to merit referee time.","headline":"This paper gives explicit Strauss and Fujita critical exponents for small-data global existence in two specific families of scale-invariant damped dispersive equations.","tokens_in":2489,"tokens_out":331,"would_cite":false,"duration_ms":18109,"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":"Global existence of small-data solutions holds for semilinear damped waves when the nonlinearity power exceeds a critical exponent that depends on the dispersion operator.","keywords":["dispersive estimates","global existence","time-dependent damping","semilinear wave equations","critical exponents","Boussinesq operator","plate operator","small data solutions"],"falsifier":"A concrete counter-example or numerical simulation showing finite-time blow-up for initial data of arbitrarily small size when α lies below the predicted α_crit for either model operator.","tokens_in":2810,"feed_emoji":"","tokens_out":774,"duration_ms":12624,"temperature":0.7,"pith_summary":"The paper establishes global-in-time existence of small solutions to the Cauchy problem for semilinear evolution equations that include a scale-invariant time-dependent damping term of the form μ/(1+t) times the velocity. The proof relies on dispersive estimates that close a fixed-point argument once the power α in the nonlinearity exceeds a threshold α_crit. Two concrete model cases are treated: Boussinesq-type dispersion combined with a derivative nonlinearity yields a Strauss-type critical exponent, while plate-type dispersion with a pure power nonlinearity yields a Fujita-type critical exponent. A sympathetic reader would care because the result gives precise conditions under which damping prevents finite-time blow-up for small initial data in these wave-type models.","feed_headline":"Global small-data solutions exist above critical power for damped waves","feed_subtitle":"Dispersive estimates close the existence proof for Boussinesq and plate operators once nonlinearity power exceeds Strauss or Fujita threshol","key_machinery":"Dispersive decay estimates for the linear evolution generated by the operator L_w² together with the precise scale-invariant damping μ/(1+t), used to close an iteration or fixed-point argument for the semilinear problem.","core_discovery":"For the Cauchy problem u_tt + L_w² u + μ/(1+t) u_t = Δ^θ f(u) with f(u) = |u|^α, small initial velocity data in suitable Sobolev spaces yield global solutions whenever α > α_crit, where α_crit is determined by the symbol w(ξ) of the spatial operator, the damping strength μ, and whether θ equals 0 or 1. In the Boussinesq case w(ξ) = √(|ξ|² + |ξ|⁴) with θ = 1 the threshold is of Strauss type; in the plate case w(ξ) = |ξ|^σ (σ ≥ 2) with θ = 0 the threshold is of Fujita type.","pith_inferences":["The critical exponents obtained here should coincide with the known thresholds for the corresponding undamped problems when the damping parameter μ is taken to zero.","Analogous global-existence statements may hold for other dispersion symbols w(ξ) whose Fourier multipliers admit similar pointwise decay estimates.","Below the critical exponent one expects blow-up even for small data, by analogy with the classical Strauss and Fujita results."],"forward_implications":["For Boussinesq operators with derivative nonlinearity, small-data solutions exist globally in time precisely when the power exceeds the Strauss critical value.","For plate operators with power nonlinearity, small-data solutions exist globally when the power exceeds the Fujita critical value.","The same linear dispersive estimates plus the scale-invariant damping close the nonlinear iteration in both settings once the power condition holds."],"fun_headline_variants":["Global small solutions above critical exponents for damped wave equations","Strauss and Fujita type critical exponents for global small data in damped waves","Boussinesq plate equations allow global small solutions above critical thresholds","Time-dependent damping leads to global small-data solutions above alpha critical"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The initial velocity must be small enough in a Sobolev norm adapted to the dispersion operator, and the damping coefficient must be exactly of the form μ/(1+t).","fun_headline_variants_meta":{"raw":{"variants":["Global small solutions above critical exponents for damped wave equations","Strauss and Fujita type critical exponents for global small data in damped waves","Boussinesq plate equations allow global small solutions above critical thresholds","Time-dependent damping leads to global small-data solutions above alpha critical"]},"model":"grok-4.3","cost_usd":0.009296,"raw_usage":{"total_tokens":4242,"prompt_tokens":832,"num_sources_used":0,"completion_tokens":70,"cost_in_usd_ticks":92962000,"prompt_tokens_details":{"text_tokens":832,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3340,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":832,"tokens_out":70,"duration_ms":24663,"temperature":1.0,"reasoning_tokens":3340,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T12:21:16.985151+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete counter-example or numerical simulation showing finite-time blow-up for initial data of arbitrarily small size when α lies below the predicted α_crit for either model operator.","supporting_citations":[],"review_version":1}