{"id":"09392f3e-970a-4a71-863f-779fab64878b","arxiv_id":"1908.02989","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"On the Heisenberg group, the semilinear damped wave equation with |u|^p has Fujita critical exponent p_Fuj(Q)=1+2/Q: global small-data solutions for p above it, blow-up for p at or below it.","lead":"The paper proves that the critical exponent for the semilinear damped wave equation with power nonlinearity on the Heisenberg group is the Fujita exponent 1 + 2/Q, where Q is the homogeneous dimension. Small data solutions exist globally above this exponent, while solutions blow up in finite time at and below it, under a standard sign condition on the initial data.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Global existence rests on linear decay estimates whose proof is deferred to companion [17]; the Fujita threshold p>1+2/Q is read off those rates, so they must be independently verified.","rationale":"I read the paper in good faith. The blow-up proof via the test function method appears internally sound: the test function estimates (55), the subcritical scaling (58), and the critical-case support refinement leading to (59) are consistent. The local existence contraction argument is standard, and the weighted Gagliardo-Nirenberg inequalities in Section 4 are derived within the manuscript. The main theorems therefore hinge on the linear decay estimates in Proposition 6.1, exactly as the reader identified. Those estimates are imported from the companion preprint [17] with no proof reproduced here. Every global existence estimate in Section 7 uses these rates, and the critical exponent condition (49) is read off them. This is a genuine, concrete dependency rather than a stylistic concern. It does not make the result implausible: the claimed rates match the Euclidean analogy and the expected heat-like behavior on stratified Lie groups, and the companion paper appears intended to supply them. But for the present manuscript to be self-contained and fully verified, those rates need independent confirmation. The reader's CONDITIONAL verdict is appropriate; no change is needed, and the concrete check above would settle the concern.","tokens_in":21797,"tokens_out":19895,"duration_ms":204840,"concrete_test":"Independently re-derive Proposition 6.1 using the group Fourier transform on the Heisenberg group, as indicated in [17]. Specifically, for the case n = 1 (Q = 4), derive the explicit symbol of the damped wave propagator and verify the sharp L1–L2 decay ||u(t)||_2 ≤ C(1+t)^{-1} for (u0,u1) ∈ (L1∩L2)^2, with ||u_t(t)||_2 ≤ C(1+t)^{-2}. If these rates fail, or if logarithmic corrections appear at low frequencies of the sub-Laplacian, recompute the threshold in (49) and the range in Theorem 2.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central global existence theorem (Theorem 2.2) is driven by Proposition 6.1, which supplies the L1–L2 and L2–L2 decay rates for the linear damped wave equation on the Heisenberg group. The proof of Proposition 6.1 is not included; it is only referenced as [17, Theorem 1.1]. This is load-bearing because every Duhamel term in the proof of Theorem 2.2 is estimated using exactly these rates: (36) gives ||u(t)||_2 ≲ (1+t)^{-Q/4}, (37) and (38) give the corresponding gradient and time-derivative rates, and (39)–(41) control the second Duhamel interval. In particular, the critical threshold p > 1 + 2/Q is obtained from the integrability condition (49), which reads -Qp/2 + Q/2 + δp(Q/4 + 1/2) < -1; this condition would change if the L1–L2 decay exponent in Proposition 6.1 were different. Thus the main claim is conditional on correctness of estimates that this manuscript does not itself establish. The paper does state the dependency explicitly, but it remains an unverified external pillar. The abstract's omission of the upper bound p ≤ p_GN(Q) is a secondary presentation issue; the load-bearing concern is the deferred linear theory.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the Cauchy problem for the semilinear damped wave equation u_tt - Δ_H u + u_t = |u|^p on the Heisenberg group H^n, with initial data in a weighted energy space. The main claim is that the critical exponent is the Fujita exponent p_Fuj(Q) = 1 + 2/Q, where Q = 2n + 2 is the homogeneous dimension. Section 2 states a local existence theorem (Theorem 2.1), a global small-data existence theorem for p_Fuj(Q) < p ≤ p_GN(Q) (Theorem 2.2), and a finite-time blow-up theorem for 1 < p ≤ p_Fuj(Q) under a sign condition on the data (Theorem 2.3). The strategy combines weighted energy identities with the weight ψ(t,η) = (|x|^2 + |y|^2 + 4|τ|)/(8(1+t)), a weighted Gagliardo-Nirenberg inequality (Lemma 4.3), Duhamel's principle with linear decay estimates (Proposition 6.1, cited from the companion paper [17]), and the test-function method for blow-up. The proof structure is standard and the estimates in the submitted text appear internally consistent, but the global existence theorem is not self-contained and the abstract overstates the range of p for which it is proved.","tokens_in":21964,"tokens_out":8989,"duration_ms":88406,"significance":"If the missing linear-theory input is supplied, the paper establishes a Fujita-type dichotomy for a semilinear damped wave equation on a non-abelian stratified Lie group, with the threshold emerging from the homogeneous dimension rather than the topological dimension. The test-function proof is self-contained and gives the sharp threshold including the critical case. The weighted Gagliardo-Nirenberg inequality and the bootstrap argument are natural extensions of the Euclidean approach. However, the claim is not fully self-contained: Theorem 2.2 is conditional on Proposition 6.1, whose proof is deferred to [17], and the abstract overstates the range of p for which global existence is proved.","major_comments":[{"comment":"The linear decay estimates (36)-(41) are not proved in this manuscript; the proof is only a reference to [17, Theorem 1.1]. These estimates are load-bearing: every Duhamel term in the proof of Theorem 2.2 is bounded using (36)-(41), and the integrability condition (49), which yields the threshold p > 1 + 2/Q, is obtained from the decay exponent -Q/4 in (36). Thus the global existence theorem is conditional on a companion preprint. The dependency is stated honestly, but for a journal paper the proof of Proposition 6.1, or at least of the rates (36)-(38), should be included or made available; otherwise the central claim is not independently verifiable.","section":"Section 6, Proposition 6.1"},{"comment":"The abstract and introduction state global existence for all p > p_Fuj(Q), but Theorem 2.2 proves it only for 1 < p ≤ p_GN(Q) = Q/(Q-2). The upper bound is used in the proof: before (48) the authors note that Lemma 4.3 applies \"thanks to the upper bound p ≤ p_GN(Q)\". Since for n = 1 (Q = 4) the proved super-Fujita range is only (3/2, 2], the abstract overstates the result. The statements should be corrected to match the theorem, or the range of p should be extended and discussed.","section":"Abstract / Theorem 2.2"}],"minor_comments":[{"comment":"The statement says u ∈ L^p_loc([0,T) × R^n), but the spatial domain should be the Heisenberg group H^n.","section":"Theorem 2.3"},{"comment":"The phrase \"where where C is a nonnegative constant\" contains a duplicated word; also, C should presumably be a positive constant.","section":"Lemma 4.1"},{"comment":"The text contains several spacing and OCR-type artifacts (e.g., \"Pon tecorvo\", \"Bonc hev\", \"lin ear\"); the manuscript should be carefully proofread before final submission.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main issue for the editor is that Theorem 2.2 is conditional on the companion preprint [17], whose proof is not included. The authors should be asked to provide the proof of Proposition 6.1 or to state the theorem conditionally. The abstract overstates the range of p, and this should be corrected. Apart from these points, the submitted proof is coherent and the test-function blow-up argument is self-contained."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a solid paper, not a repackaging. It proves that the semilinear damped wave equation on the Heisenberg group has the expected Fujita exponent: global small-data solutions for p_Fuj < p ≤ p_GN(Q), and finite-time blow-up for 1 < p ≤ p_Fuj under the sign condition. The Euclidean analogue was known; the Heisenberg case is genuinely new.\n\nWhat the paper does well: the weighted energy identity (15), the inequality (21), and the weighted Gagliardo-Nirenberg inequality of Lemma 4.3 are consistent and worked out in detail. The blow-up proof in Section 8 is self-contained and handles the critical case with a refined cut-off argument; I checked that part and it holds up. The critical exponent is not put in by hand—it emerges naturally from the decay rates and integrability conditions.\n\nThe real soft spot is Proposition 6.1. The L1–L2 and L2–L2 decay estimates are imported from the companion preprint [17] without proof. These rates control every Duhamel term in Section 7, and the threshold p > 1+2/Q is read off exactly those exponents. If the rates in [17] shift, the global existence result shifts. The paper explicitly states the dependency, which is honest, but it is still an unverified external pillar. For a definitive version, either [17] should be available and refereed, or the key estimates should be sketched in an appendix.\n\nSecondary issues: the abstract overclaims by omitting the upper bound p ≤ p_GN(Q) required in Theorem 2.2. That is a presentation slip, not a mathematical error. There are also minor typos—Theorem 2.3 states the solution space as L^p_loc([0,T) × R^n) but it should be H^n, and the exponent near (58) is garbled in places. All fixable.\n\nBottom line: the central theorem appears correct, and the extension of the weighted-energy method to the Heisenberg group is nontrivial and well executed. The blow-up half can stand alone; the global-existence half should be accepted only with the companion paper's estimates verified or included. I would send this to a serious referee rather than desk-reject, with that condition attached.","headline":"Genuinely new Fujita-type result on the Heisenberg group with a sound blow-up proof, but the global-existence half rests on linear decay estimates deferred to a companion preprint.","tokens_in":22633,"tokens_out":2883,"would_cite":true,"duration_ms":27827,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B33","35L71","35R03","35B44","35B45","43A80","58J45"],"pacs":[],"model":"deepseek-v4-flash","headline":"On the Heisenberg group, the damped wave equation with power nonlinearity has the Fujita critical exponent 1 + 2/Q.","keywords":["damped wave equation","Heisenberg group","sub-Laplacian","Fujita exponent","critical exponent","blow-up","test function method","exponentially weighted energy spaces"],"falsifier":"Check the linear decay estimate directly for the homogeneous damped wave equation on the Heisenberg group with initial data in $L^1 \\cap L^2$: test whether $(1+t)^{Q/4}\\|u(t)\\|_{L^2}$ remains bounded. If the true decay exponent is $\\alpha < Q/4$, then the integrability argument that produces the threshold $p > 1 + 2/Q$ breaks down and the claimed critical exponent would not follow from this method; alternatively, exhibiting a global solution for $p = 1 + 2/Q$ with data satisfying (9) would refute the blow-up result.","tokens_in":21449,"feed_emoji":"💥","tokens_out":8645,"duration_ms":81363,"temperature":0.7,"pith_summary":"The paper studies the semilinear damped wave equation $u_{tt} - \\Delta_{\\mathbb{H}} u + u_t = |u|^p$ on the Heisenberg group $\\mathbb{H}^n$, where $\\Delta_{\\mathbb{H}}$ is the sub-Laplacian and the group has homogeneous dimension $Q = 2n + 2$. It claims that the critical exponent separating global existence from finite-time blow-up is the Fujita exponent $p_{\\mathrm{Fuj}}(Q) = 1 + 2/Q$, exactly as in Euclidean space with $Q$ in place of the spatial dimension. For $p$ above this threshold, small initial data in an exponentially weighted energy space produce global solutions with explicit decay rates; for $1 < p \\leq p_{\\mathrm{Fuj}}(Q)$, solutions whose data have sign-definite averages blow up in finite time. This matters because it shows the Fujita threshold is governed by the homogeneous dimension of the ambient group rather than by the noncommutative details of its sub-Riemannian geometry.","feed_headline":"Critical Fujita exponent on the Heisenberg group is 1 + 2/Q","feed_subtitle":"Above the threshold small data yield global solutions; at or below it, solutions blow up in finite time.","key_machinery":"The load-bearing objects are the Heisenberg group $\\mathbb{H}^n$ with homogeneous dimension $Q = 2n + 2$ and the time-dependent exponential weight $\\psi(t,\\eta) = (|x|^2 + |y|^2 + 4|\\tau|)/(8(1+t))$. The weight converts the sub-Laplacian into a Schr\\\"odinger-type operator with a potential that yields a weighted energy identity: multiplying the equation by $e^{2\\psi} u_t$ produces a divergence-plus-time-derivative structure with controllable remainders, and $\\psi_t \\leq 0$ supplies the sign needed to bound the nonlinearity. On the existence side, Proposition 6.1 supplies the linear decay estimates $\\|u(t)\\|_{L^2} \\lesssim (1+t)^{-Q/4}$ for $L^1$ data, and Lemma 4.3, the weighted Gagliardo-Nirenberg inequality, translates $L^2$-decay into the $L^{2p}$-estimates needed to close Duhamel's integral; the integrability condition that yields $p > 1 + 2/Q$ is read off these decay exponents. On the blow-up side, the test functions $\\phi_R(t,x,y,\\tau) = \\beta(t/R^2) \\alpha(x/R) \\alpha(y/R) \\beta(\\tau/R^2)$ localize in boxes $D_R$ of measure $\\approx R^Q$, and H\\\"older's inequality forces $I_R \\to 0$ for $p < p_{\\mathrm{Fuj}}(Q)$, while the critical case $p = p_{\\mathrm{Fuj}}(Q)$ is handled by refining the support analysis.","core_discovery":"On its own terms, the paper's central discovery is a complete Fujita-type dichotomy for the Heisenberg damped wave equation. Theorem 2.3 states that when $1 < p \\leq p_{\\mathrm{Fuj}}(Q)$ and the Cauchy data satisfy $\\liminf_{R \\to \\infty} \\int_{D_R} (u_0 + u_1) \\, d\\eta > 0$, every local weak solution blows up in finite time, no matter how small the data are. Theorem 2.2 states the complementary half: for $p_{\\mathrm{Fuj}}(Q) < p \\leq p_{\\mathrm{GN}}(Q) = Q/(Q-2)$, there is $\\varepsilon_0 > 0$ such that data of weighted-energy norm at most $\\varepsilon_0$ admit a unique global solution $u$ with $\\|u(t)\\|_{L^2} \\lesssim (1+t)^{-Q/4}$, $\\|\\nabla_{\\mathbb{H}} u(t)\\|_{L^2} \\lesssim (1+t)^{-Q/4 - 1/2}$, and $\\|u_t(t)\\|_{L^2} \\lesssim (1+t)^{-Q/4 - 1}$. The blow-up proof uses the test function method with scaled bump functions; the existence proof uses Duhamel's principle, decay estimates for the linear equation, weighted Gagliardo-Nirenberg inequalities, and a contraction argument in exponentially weighted energy spaces.","pith_inferences":["The same test-function argument should go through verbatim for other stratified Lie groups whose homogeneous dimension is $Q$, since only the measure of the boxes $D_R$ and the scaling of the sub-Laplacian enter; the blow-up half of the dichotomy is thus likely generic, while the existence half will track whatever linear decay estimates are available.","If the companion preprint's linear rates are optimal, then the critical exponent here is optimal too, and lifespan upper bounds for $p \\leq 1 + 2/Q$ would be the natural next project, matching the known Euclidean picture.","The exponential weight $\\psi(t,\\cdot)$ is tailored to the heat-like scaling of the damped wave equation; a similar weight should work for the corresponding semilinear heat equation on the Heisenberg group, connecting the two Fujita results.","A testable extension of the paper's reasoning: replace $|u|^p$ by $|u|^p$ with an additional time factor $t^\\beta$, and check whether the threshold shifts to $1 + (2+\\beta)/Q$, as it does in Euclidean models."],"forward_implications":["Global small-data solutions exist exactly when $p > 1 + 2/Q$, with the additional technical ceiling $p \\leq Q/(Q-2)$; the decay rates of the linear equation are preserved by the nonlinear problem.","For every $1 < p \\leq 1 + 2/Q$, any weak solution whose data have positive limiting average over the boxes $D_R$ blows up in finite time, including at the critical value $p = 1 + 2/Q$.","The threshold is determined only by the homogeneous dimension $Q = 2n + 2$, not by the noncommutativity or the step of the group.","The weighted energy space $A(\\mathbb{H}^n)$ is sufficient to absorb the nonlinearity: the $L^2$, gradient, and time-derivative norms all decay at the same rates as the Euclidean case with dimension $Q$.","No critical-case global existence occurs under the sign assumption (9); the lifespan at $p = 1 + 2/Q$ is always finite for such data."],"supporting_citations":[{"why":"supplies the $L^1$-$L^2$ and $L^2$-$L^2$ decay estimates for the linear damped wave equation on $\\mathbb{H}^n$ that every Duhamel estimate in the global-existence proof uses","marker":"[17]"},{"why":"established the Euclidean critical exponent via exponentially weighted energy spaces, the strategy adapted here","marker":"[21]"},{"why":"proved the Euclidean global-existence result without compact support assumptions, the source of the exponential-weight approach","marker":"[12]"},{"why":"derived the $L^2$ decay estimates for the linear damped wave equation in Euclidean space that [17] generalizes","marker":"[13]"},{"why":"determined the Fujita exponent for semilinear heat equations on sub-Riemannian manifolds and unimodular Lie groups, setting the expected threshold","marker":"[20]"},{"why":"used the test function method for the semilinear heat equation on the Heisenberg group, the template for the blow-up proof here","marker":"[10]"},{"why":"extends the test-function blow-up method to stratified Lie groups, covering the sub-Fujita range","marker":"[11]"},{"why":"is the test function method reference from which the blow-up technique is drawn","marker":"[14]"},{"why":"provides the Gagliardo-Nirenberg type inequalities and the graded-Lie-group damped wave model that motivate the interpolation arguments","marker":"[19]"}],"fun_headline_variants":["Heisenberg damped wave: Fujita exponent 1 + 2/Q","Critical exponent for Heisenberg damped wave equations","Fujita threshold on Heisenberg group: global vs blow-up","Small data global, large blow-up: Heisenberg Fujita exponent","Heisenberg wave blow-up threshold: p = 1 + 2/Q"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The global-existence half depends on the linear decay estimates of Proposition 6.1, which are taken from a companion preprint rather than proved here; if those rates—for instance $\\|u(t)\\|_{L^2} \\lesssim (1+t)^{-Q/4}$ for $L^1$ data—were different, the claimed threshold $p > 1 + 2/Q$ would not be forced by the Duhamel argument.","fun_headline_variants_meta":{"raw":{"variants":["Heisenberg damped wave: Fujita exponent 1 + 2/Q","Critical exponent for Heisenberg damped wave equations","Fujita threshold on Heisenberg group: global vs blow-up","Small data global, large blow-up: Heisenberg Fujita exponent","Heisenberg wave blow-up threshold: p = 1 + 2/Q"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000224,"raw_usage":{"total_tokens":1475,"prompt_tokens":976,"completion_tokens":499,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":592,"completion_tokens_details":{"reasoning_tokens":408}},"tokens_in":592,"tokens_out":499,"duration_ms":5296,"temperature":1.0,"reasoning_tokens":408,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:29:29.627683+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the linear decay estimate directly for the homogeneous damped wave equation on the Heisenberg group with initial data in $L^1 \\cap L^2$: test whether $(1+t)^{Q/4}\\|u(t)\\|_{L^2}$ remains bounded. If the true decay exponent is $\\alpha < Q/4$, then the integrability argument that produces the threshold $p > 1 + 2/Q$ breaks down and the claimed critical exponent would not follow from this method; alternatively, exhibiting a global solution for $p = 1 + 2/Q$ with data satisfying (9) would refute the blow-up result.","supporting_citations":[{"cited_title":"Decay estimates for the linear damped wave equation on the Heisenberg group","cited_arxiv_id":"1908.02657","evidence_quote":"supplies the $L^1$-$L^2$ and $L^2$-$L^2$ decay estimates for the linear damped wave equation on $\\mathbb{H}^n$ that every Duhamel estimate in the global-existence proof uses"},{"cited_title":"Differential Equations 174(2) (2001), 464-489","cited_arxiv_id":null,"evidence_quote":"established the Euclidean critical exponent via exponentially weighted energy spaces, the strategy adapted here"},{"cited_title":"61(7) (2005), 1189-1208","cited_arxiv_id":null,"evidence_quote":"proved the Euclidean global-existence result without compact support assumptions, the source of the exponential-weight approach"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"derived the $L^2$ decay estimates for the linear damped wave equation in Euclidean space that [17] generalizes"},{"cited_title":"Existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds, and Fujita exponent on unimodular Lie groups","cited_arxiv_id":"1812.01933","evidence_quote":"determined the Fujita exponent for semilinear heat equations on sub-Riemannian manifolds and unimodular Lie groups, setting the expected threshold"},{"cited_title":"Lifespan estimates for local in time solutions to the semilinear heat equation on the Heisenberg group","cited_arxiv_id":"1905.05696","evidence_quote":"used the test function method for the semilinear heat equation on the Heisenberg group, the template for the blow-up proof here"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"extends the test-function blow-up method to stratified Lie groups, covering the sub-Fujita range"},{"cited_title":"Steklov Inst","cited_arxiv_id":null,"evidence_quote":"is the test function method reference from which the blow-up technique is drawn"},{"cited_title":"Differential Equations 265 (2018), 5212-5236","cited_arxiv_id":null,"evidence_quote":"provides the Gagliardo-Nirenberg type inequalities and the graded-Lie-group damped wave model that motivate the interpolation arguments"}],"review_version":1}