{"id":"5b1a062f-9d15-4ee8-8b12-69df6fe45cbe","arxiv_id":"2502.13150","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For u_t = Δu + h(t)u^q on infinite graphs with λ1(G)>0, the paper claims blow-up for fast-growing h and global small data when ∫ h(t)e^{-λ1(q-1)t}dt is finite, but the general blow-up theorem is vitiated by a sign error.","lead":"The paper studies global existence versus finite-time blow-up for u_t = Δu + h(t)u^q on infinite graphs with a positive spectral gap. Its main blow-up condition contains a sign error that makes the stated theorem contradict its own global-existence result, so the paper as written is not sound.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.2's blow-up hypothesis has a sign error: combining (4.1) and (4.4) yields e^{-(λ1+ε)t}, so the condition should use e^{-(λ1+ε)t}, not e^{[λ1+ε]t}; as written it contradicts Theorem 3.4 for h(t)=e^{αt} with 0<α<(q−1)λ1.","rationale":"I read the paper as an attempt to establish a Fujita-type threshold for semilinear parabolic equations on infinite graphs with λ1(G)>0, with the threshold for h(t)=e^{αt} being α=(q−1)λ1(G). The central claim requires Theorem 3.2 and Theorem 3.4 to be consistent. The sign error in (3.2)/(4.7) is unambiguous and more load-bearing than the provenance of Proposition 2.4: even if the uniform heat-kernel bound (2.7) were fully justified, Theorem 3.2 as stated contradicts Theorem 3.4 for every positive α below the claimed threshold. The local-existence contraction argument in Section 5 and the global-existence fixed-point framework in Section 6 look plausible in themselves, but they cannot rescue the main nonexistence result. The reader's rationale already identifies the sign error, and I agree with the REJECT verdict. I mark agreement as partial rather than full because the reader's formal 'weakest_assumption' field points to Proposition 2.4, whereas I regard the sign error in the central blow-up theorem as the single load-bearing defect; the heat-kernel bound is a secondary concern about an unverified citation and does not need to be resolved to see the contradiction. The paper contains valuable ideas and may be repairable by changing (3.2) to lim H^{1/(q−1)}e^{-(λ1+ε)t}=+∞ and rechecking the resulting threshold, but as submitted the central claim is internally inconsistent.","tokens_in":11585,"tokens_out":6199,"duration_ms":57918,"concrete_test":"Recompute equation (4.7) from the two displayed inequalities in the proof of Theorem 3.2: substitute Φ_x(0) ≥ C1 e^{-(λ1+ε)T} from (4.1) and Φ_x(0) ≤ C H(T)^{-1/(q−1)} from (4.4). The correct rearrangement is H(T)^{1/(q−1)} e^{-(λ1+ε)T} ≤ C/C1, not H(T)^{1/(q−1)} e^{(λ1+ε)T} ≤ C/C1 as printed. This single algebra check settles the sign issue. For a concrete contradiction, set h(t)=e^{αt} with α=(q−1)λ1/2: the printed (3.2) holds because α/(q−1)+λ1+ε>0, while (3.3) also holds because α−(q−1)λ1<0, so Theorems 3.2 and 3.4 cannot both be true. No numerical simulation is needed; the contradiction is immediate from the two stated theorems plus this choice of h.","verdict_should_be":"REJECT","load_bearing_attack":"The load-bearing assertion is the sharp dichotomy in Theorem 3.2/Remark 3.5: for h(t)=e^{αt}, any nontrivial solution blows up when α>(q−1)λ1, while Theorem 3.4 gives global small-data solutions when α<(q−1)λ1. As written, the blow-up hypothesis (3.2) has the wrong exponential sign. In the proof, Lemma 4.1 gives Φ_x(0) ≥ C1 e^{-(λ1+ε)T} and Lemma 4.2 gives Φ_x(0) ≤ C H(T)^{-1/(q−1)}. Combining these and rearranging yields H(T)^{1/(q−1)} e^{-(λ1+ε)T} ≤ C/C1. A contradiction requires this quantity to tend to +∞, so the hypothesis should be lim H^{1/(q−1)}e^{-(λ1+ε)t}=+∞. The manuscript instead states (3.2) and (4.7) with e^{[λ1+ε]T}, i.e. the opposite sign. With the stated sign, take h(t)=e^{αt} and any 0<α<(q−1)λ1. Then the exponent α/(q−1)+λ1+ε is positive, so (3.2) holds and Theorem 3.2 predicts finite-time blow-up for every nontrivial initial datum. But this same parameter range is exactly where (3.3) holds, so Theorem 3.4 produces a nontrivial global solution for suitably small u0. The two theorems are therefore contradictory in an open parameter range; the central threshold claim is internally inconsistent as stated. The intended threshold may be recoverable by correcting the sign, but that changes the theorem's hypothesis and the proof's contradiction argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the semilinear parabolic equation u_t = Δu + h(t)u^q on infinite weighted graphs with λ1(G) > 0. The main results are a finite-time blow-up theorem (Theorem 3.2) under a growth condition on H(t) = ∫ h, a local existence theorem (Theorem 3.3), and a global existence theorem (Theorem 3.4) for small initial data when ∫ h(t)e^{-λ1(q-1)t} dt < ∞. For the model case h(t) = e^{αt}, the authors claim a sharp threshold: blow-up for α > (q−1)λ1 and global existence for α < (q−1)λ1. The proofs use heat-kernel estimates, a Jensen-inequality argument, and contraction mappings in a heat-kernel-weighted metric space.","tokens_in":11993,"tokens_out":5461,"duration_ms":49891,"significance":"The intended result is a meaningful extension of Fujita-type thresholds from hyperbolic space to graphs with a spectral gap, and the contraction method in a heat-kernel-weighted space is a useful technical contribution. However, the main blow-up theorem as stated is internally inconsistent with the global existence theorem, so the significance can only be assessed after the sign error in (3.2) and (4.7) is corrected and the provenance of the uniform heat-kernel bound (2.7) is clarified.","major_comments":[{"comment":"The hypothesis of Theorem 3.2 has the wrong sign in the exponential. Combining Lemma 4.1 (Eq. (4.1)) and Lemma 4.2 (Eq. (4.4)) gives C1 e^{-(λ1+ε)T} ≤ Φ(0) ≤ (1/(q−1))^{1/(q−1)} H(T)^{-1/(q−1)}, so rearrangement yields H(T)^{1/(q−1)} e^{-(λ1+ε)T} ≤ constant. A contradiction requires H(T)^{1/(q−1)} e^{-(λ1+ε)T} → ∞, i.e., hypothesis (3.2) should have e^{-[λ1+ε]t} instead of e^{[λ1+ε]t}. As written, (3.2) is satisfied for h(t)=e^{αt} with any α>0, including 0<α<(q−1)λ1, where Theorem 3.4 guarantees a global small-data solution because (3.3) holds. The two theorems therefore contradict each other in an open parameter range. The intended threshold α>(q−1)λ1 is recovered after the sign correction.","section":"Theorem 3.2, Eq. (3.2), proof (4.7)"},{"comment":"The statement 'by combining together [9, Theorems 2.1, 2.2]' is not reliable because reference [9] is Fujita's 1966 paper on blow-up for u_t=Δu+u^{1+α}, which does not address graph heat kernels. The uniform bound p(x,y,t) ≤ C e^{-λ1(G)t} in (2.7) is used essentially in Lemma 6.2 and Proposition 6.3 to control u^{q−1}, so its validity for the class of weighted graphs considered must be established or correctly referenced (e.g., [4], [8], [10], or [26]).","section":"Proposition 2.4"}],"minor_comments":[{"comment":"The letter 'X' is used instead of 'G' in several places (e.g., 'for all x ∈ X', 'y ∈ X', 'G×(0,T)'), which is confusing and should be made uniform.","section":"Section 5 (Lemmas 5.1, 5.2 and proof of Theorem 3.3)"},{"comment":"The expression 'e^{-λ0(q−1)s}' should read 'e^{-λ1(G)(q−1)s}'.","section":"Proposition 6.3, last line"},{"comment":"The notation 'for any t>0 ... for all t ≥ t' overloads t; a different symbol, such as t0, should be used for the threshold time.","section":"Proposition 2.4"},{"comment":"The name 'Caccioppoli' is misspelled as 'Cacioppoli' in both occurrences of the Banach-Caccioppoli theorem.","section":"Proofs of Theorems 3.3 and 3.4"}],"recommendation":"major_revision","confidential_remarks":"The sign error in the main blow-up hypothesis and the misattributed heat-kernel bound are both load-bearing. I recommend that the authors correct the sign, supply a proper reference or proof for Proposition 2.4, and carefully audit the manuscript for similar citation errors before the paper is reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper is not publishable in its current form. The headline result, Theorem 3.2, states a blow-up condition with e^{+[λ1+ε]t}, but the proof (combining Lemmas 4.1 and 4.2) actually yields H^{1/(q-1)} e^{-[λ1+ε]t} ≤ const. That makes the theorem false as stated: for h(t)=e^{αt} with 0<α<(q-1)λ1, the hypothesis (3.2) is satisfied, so Theorem 3.2 predicts finite-time blow-up for every nontrivial u0, while Theorem 3.4 gives a global small-data solution in exactly that parameter range. These two theorems directly contradict each other in an open set of parameters.\n\nWhat is genuinely new: this is the first paper I know that treats time-dependent multipliers h(t) in the source u^q on infinite graphs with λ1>0, and the contraction argument in the heat-kernel-weighted metric (Section 6) is a real technical adaptation, not a copy of the manifold proofs. The local existence part is clean. If the sign is fixed, the intended dichotomy (blow-up for α>(q-1)λ1, global for α<(q-1)λ1) would follow from the same strategy.\n\nThe other soft spot is Proposition 2.4: the uniform heat-kernel bound p(x,y,t)≤C e^{-λ1 t} is cited to [9], which is Fujita's 1966 blow-up paper—not a heat-kernel reference on graphs. That bound is load-bearing for the global existence proof (Lemmas 6.2 and 6.3). It may be true under hypotheses like stochastic completeness plus something else, but the paper does not say what, and the citation is simply wrong. This needs to be fixed before the global existence half can be trusted.\n\nThere are also smaller issues: the abstract says 'global in time existence or finite time blow-up' but the critical case α=(q-1)λ1 is left open; and Remark 3.6's h≡1 statement is already in [16], so the only new content is the time-dependent h.\n\nBottom line: the intended result is likely correct and the paper would be a reasonable contribution to the graph-parabolic literature after a major revision. But as written, the central theorem is false and the proof is internally inconsistent. I would not cite it in its current form. If you're the editor, I'd send it to a referee—someone who can check the sign fix and press on the heat-kernel bound—rather than desk-reject, because the core idea is salvageable.","headline":"The paper's main blow-up theorem has an exponential sign error in its hypothesis, making it internally inconsistent with its own global existence result; the intended Fujita-type threshold on graphs is plausible but needs a major revision.","tokens_in":12515,"tokens_out":5619,"would_cite":false,"duration_ms":53180,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A01","35A02","35B44","35K05","35K58","35R02"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a spectral-gap threshold for finite-time blow-up versus global existence of solutions to $u_t=\\Delta u+h(t)u^q$ on infinite weighted graphs.","keywords":["semilinear parabolic equation","infinite weighted graphs","finite-time blow-up","global existence","heat kernel","spectral gap","time-dependent source"],"falsifier":"Construct a stochastically complete infinite weighted graph with $\\lambda_1(G)>0$ whose heat kernel can be computed or sharply bounded, and check whether $\\sup_{x,y}p(x,y,t)e^{\\lambda_1(G)t}$ remains bounded as $t\\to\\infty$. If it is unbounded, the uniform estimate (2.7) behind the global-existence proof fails; if it stays bounded, the paper's global-existence mechanism has the estimate it needs.","tokens_in":11397,"feed_emoji":"🔥","tokens_out":16892,"duration_ms":165537,"temperature":0.7,"pith_summary":"The paper studies the Cauchy problem $u_t=\\Delta u+h(t)u^q$, $q>1$, on infinite weighted graphs, and asks whether solutions exist for all time or blow up in finite time. It claims that when the graph Laplacian has positive spectral bottom $\\lambda_1(G)$, the answer is governed by the competition between the growth of $h$ and the decay rate $\\lambda_1(G)$ of the heat semigroup: for $h(t)=e^{\\alpha t}$, every nontrivial nonnegative solution blows up in finite time if $\\alpha>(q-1)\\lambda_1(G)$, while small data admit a global solution if $\\alpha<(q-1)\\lambda_1(G)$. This is the graph analogue of a known hyperbolic-space dichotomy, and it matters because it identifies the spectral gap, not volume growth alone, as the quantity controlling blow-up for time-dependent sources. The critical equality case is left open.","feed_headline":"Blow-up on infinite graphs follows a spectral-gap rule","feed_subtitle":"All solutions blow up when the source outruns (q−1) times the lowest Laplacian eigenvalue; small data survive.","key_machinery":"The argument turns on the heat kernel $p(x,y,t)$ of the graph and the quantity $\\lambda_1(G)$, the bottom of the $L^2$ spectrum of $-\\Delta$. Two estimates do the work. On the blow-up side, the exponential-time asymptotic of the heat kernel gives $e^{t\\Delta}u_0(x_0)\\ge C_1e^{-[\\lambda_1(G)+\\varepsilon]t}$ for large $t$, and a Jensen-type integration on the weighted average $\\Phi_x(t)=\\sum_z p(x,z,T-t)u(z,t)\\mu(z)$ yields $(q-1)H(T)[\\Phi_x(0)]^{q-1}\\le1$; comparing the two forces the contradiction when $H(T)^{1/(q-1)}$ outgrows $e^{[\\lambda_1(G)+\\varepsilon]T}$. On the global-existence side, the fixed point is placed in the complete metric space of functions bounded by multiples of $p(x,y_0,t+\\gamma)$, and the contraction is small because the uniform exponential bound $p(x,y,t)\\le Ce^{-\\lambda_1(G)t}$ turns $u^{q-1}\\le M^{q-1}p(\\cdot,y_0,\\cdot+\\gamma)^{q-1}$ into an integrable factor $\\delta^{q-1}e^{-\\lambda_1(G)(q-1)s}h(s)$.","core_discovery":"The central discovery, on the paper's own terms, is a dichotomy for the semilinear heat equation on stochastically complete infinite weighted graphs with $\\lambda_1(G)>0$. Theorem 3.2 says that if $H(t)=\\int_0^t h(s)\\,ds$ satisfies $H(t)^{1/(q-1)}e^{[\\lambda_1(G)+\\varepsilon]t}\\to+\\infty$ for some $\\varepsilon\\in(0,\\lambda_1(G))$, then no nontrivial nonnegative solution can be global; blow-up occurs in finite time. Theorem 3.4 says that if $\\int_0^\\infty h(t)e^{-\\lambda_1(G)(q-1)t}\\,dt<\\infty$ and the initial datum is smaller than a small multiple of a heat kernel $p(\\cdot,y_0,\\gamma)$, then a global mild solution exists and stays under $M p(\\cdot,y_0,t+\\gamma)$. For $h(t)=e^{\\alpha t}$ the two theorems combine into $\\alpha>(q-1)\\lambda_1(G)$ for universal finite-time blow-up and $\\alpha<(q-1)\\lambda_1(G)$ for small-data global existence, leaving only $\\alpha=(q-1)\\lambda_1(G)$ unresolved.","pith_inferences":["Taken literally, condition (3.2) appears to carry the wrong sign: for $h(t)=e^{\\alpha t}$, the displayed limit is $+\\infty$ for every $\\alpha>0$, which overlaps the global-existence range of Theorem 3.4; the proof's contradiction step requires $H(t)^{1/(q-1)}e^{-[\\lambda_1(G)+\\varepsilon]t}\\to+\\infty$ instead.","The same threshold should persist for more general growing sources such as $h(t)=t^\\beta e^{\\alpha t}$, with polynomial factors changing the behavior only at the critical value $\\alpha=(q-1)\\lambda_1(G)$.","Settling the open critical case on graphs will likely require sub-exponential corrections to the heat kernel, by analogy with the hyperbolic-space treatment that the paper cites for that case."],"forward_implications":["For any stochastically complete infinite graph with $\\lambda_1(G)>0$ and $h(t)=e^{\\alpha t}$ with $\\alpha>(q-1)\\lambda_1(G)$, every nontrivial nonnegative solution of the Cauchy problem blows up in finite time.","For $\\alpha<(q-1)\\lambda_1(G)$, sufficiently small initial data, pointwise no larger than a small heat-kernel bump, produce a global solution that stays under a moving heat kernel.","With $h\\equiv1$, the theorem gives global small-data solutions for every $q>1$, so the new phenomenon introduced by the paper is the time-dependent driving term and its competition with the spectral gap.","The criterion $\\int_0^\\infty h(t)e^{-\\lambda_1(G)(q-1)t}dt<\\infty$ provides a checkable sufficient condition for global existence on any graph with a spectral gap."],"supporting_citations":[{"why":"Supplies the hyperbolic-space problem whose blow-up strategy and threshold the paper adapts to graphs.","marker":"[3]"},{"why":"Cited as the source of the uniform exponential heat-kernel estimate (2.7) that carries the global-existence contraction argument.","marker":"[9]"},{"why":"Provides the prior infinite-graph blow-up and global-existence results for $h\\equiv1$ and the fixed-point framework the paper modifies for time-dependent $h$.","marker":"[16]"},{"why":"Monograph that supplies the heat kernel, the spectral-bottom asymptotic, and the positivity-improving property used throughout.","marker":"[26]"}],"fun_headline_variants":["Spectral gap sets the blow-up threshold on graphs","Graph heat blows up when source outruns spectral gap","Small data survive on graphs if source grows slowly","Infinite graph heat: source growth vs spectral gap decides","Blow-up threshold on graph heat: (q−1) times spectral gap"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The global-existence half rests on the assumption that on every graph with $\\lambda_1(G)>0$ the heat kernel decays uniformly in space, $p(x,y,t)\\le Ce^{-\\lambda_1(G)t}$ for all $x,y$ and all large $t$; if some spectral-gap graph violates this uniform bound, the contraction argument has no basis.","fun_headline_variants_meta":{"raw":{"variants":["Spectral gap sets the blow-up threshold on graphs","Graph heat blows up when source outruns spectral gap","Small data survive on graphs if source grows slowly","Infinite graph heat: source growth vs spectral gap decides","Blow-up threshold on graph heat: (q−1) times spectral gap"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000344,"raw_usage":{"total_tokens":1857,"prompt_tokens":883,"completion_tokens":974,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":499,"completion_tokens_details":{"reasoning_tokens":891}},"tokens_in":499,"tokens_out":974,"duration_ms":11462,"temperature":1.0,"reasoning_tokens":891,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T05:24:14.184455+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a stochastically complete infinite weighted graph with $\\lambda_1(G)>0$ whose heat kernel can be computed or sharply bounded, and check whether $\\sup_{x,y}p(x,y,t)e^{\\lambda_1(G)t}$ remains bounded as $t\\to\\infty$. If it is unbounded, the uniform estimate (2.7) behind the global-existence proof fails; if it stays bounded, the paper's global-existence mechanism has the estimate it needs.","supporting_citations":[{"cited_title":"Bandle, M.A","cited_arxiv_id":null,"evidence_quote":"Supplies the hyperbolic-space problem whose blow-up strategy and threshold the paper adapts to graphs."},{"cited_title":"Fujita, On the blowing up of solutions of the Cauchy problem for ut = ∆ u + u1+α , J","cited_arxiv_id":null,"evidence_quote":"Cited as the source of the uniform exponential heat-kernel estimate (2.7) that carries the global-existence contraction argument."},{"cited_title":"Graphs and Dis crete Dirichlet Spaces","cited_arxiv_id":null,"evidence_quote":"Monograph that supplies the heat kernel, the spectral-bottom asymptotic, and the positivity-improving property used throughout."}],"review_version":1}