{"id":"bbc01518-82c8-4c88-acdc-f1694c25abb6","arxiv_id":"2507.12303","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"For parabolic p-Laplacian inequalities on locally finite graphs, sufficiently fast growing nonlinear sources force finite-time blow-up of solutions.","lead":"This paper proves that solutions to a parabolic p-Laplacian inequality on infinite graphs blow up in finite time when the nonlinear source grows faster than linearly. It builds a comparison principle on finite subgraphs and compares solutions with explicitly constructed blow-up subsolutions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Missing uniform lower bound on σ makes Theorem 1.1 false: constant solutions with exponentially decaying σ are global and satisfy (1.4).","rationale":"The reader's weakest-assumption analysis correctly identifies the dependence of C0 on the unknown lifespan and the absence of a uniform lower bound on σ as the load-bearing weakness. My stress test goes further: this is not merely a gap in the proof but a counterexample to the statement as written. A spatially constant solution on any locally finite graph reduces (1.3) to the ODE y′ = σ(t)Φ(y) because the p-Laplacian annihilates constants. With integrably decaying σ(t) = e^{−t} and Φ(s) = 10s^2 (p = 3), sufficiently small constant initial data give an explicit global solution y(t) = (y0^{-1} − 10(1 − e^{−t}))^{-1}, so no blow-up occurs. The growth condition (1.4) is satisfied with, for example, C0 = 9 and ε0 = 0.5. This directly contradicts the theorem's claim that every nonnegative nontrivial initial datum produces finite-time blow-up. The only reading that avoids contradiction is that C0 is required to be δ^{-1} for a uniform δ, but that hypothesis is neither stated nor defined in the paper; the proof instead chooses δ depending on the unknown interval [0,T′]. The fix is straightforward — add a global positive lower bound on σ over the relevant finite vertex set and set C0 accordingly — but as written the central claim is false or at best ill-posed. A conditional acceptance would leave a false statement in the record; the appropriate verdict is REJECT of the current version, with the path to revision clearly identified.","tokens_in":11530,"tokens_out":19588,"duration_ms":225433,"concrete_test":"Verify the constant-solution counterexample: on G = Z take p = 3, σ(x,t) = e^{−t}, Φ(s) = 10s^2, u0(x) ≡ 0.01. Then u(x,t) = (100 − 10(1 − e^{−t}))^{-1} is a global solution of (1.3), since the p-Laplacian of a constant is zero and y′ = 10e^{−t}y^2. It satisfies (1.4) with C0 = 9, ε0 = 0.5. If this computation is correct, Theorem 1.1 as stated is false; the necessary repair is to add the hypothesis inf_{U×[0,∞)} σ ≥ δ0 > 0 and to define C0 = δ0^{-1}. Rerunning Lemma 3.1 under that added assumption should make the blow-up estimate close.","verdict_should_be":"REJECT","load_bearing_attack":"The central claim of Theorem 1.1 is not supported as stated. In Lemma 3.1 the proof chooses δ = min_{U×[0,T′]} σ and sets C0 = δ^{-1}, but T′ is the unknown existence time of the solution. For a solution that exists globally, T′ can be arbitrarily large, and no fixed positive C0 can make δΦ(s) − s^{p−1} > 0 for all t unless σ has a uniform-in-time lower bound on U. Without such a bound the argument does not close, and more seriously the statement is false. Take p = 3, G = Z with standard weights, σ(x,t) = e^{−t}, Φ(s) = 10s^2, and u0(x) ≡ y0 with 0 < y0 < 0.1. Since the p-Laplacian of a spatially constant function is zero, u(x,t) = y(t) solves (1.3) iff y′ = 10e^{−t}y^2. The explicit solution is y(t) = (y0^{-1} − 10(1 − e^{−t}))^{-1}, which is finite for all t, so there is no finite-time blow-up. All hypotheses (i)–(iv) hold, and (1.4) holds with C0 = 9 and ε0 = 0.5 because 10 ≥ 9.5. Thus the proof's C0 = δ^{-1} cannot be formed, and any fixed C0 satisfying (1.4) still admits a global non-blow-up solution. The theorem therefore needs an explicit uniform lower bound on σ over the relevant vertex set for all time.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the parabolic p-Laplacian inequality (1.3) on locally finite connected weighted graphs. It proves local existence via a Banach fixed-point argument, establishes a comparison principle on finite vertex subsets, and then uses subsolutions built from ODEs and the first Dirichlet eigenfunction to claim finite-time blow-up: Theorem 1.1 for Φ growing at least like s^{p-1}, and Theorem 1.2 for Φ ≳ s^q with 1 < q ≤ p-1 and sufficiently large initial data. The main tools are comparison with solutions on finite subsets and an eigenfunction-based Bernoulli subsolution.","tokens_in":11852,"tokens_out":16823,"duration_ms":187353,"significance":"Blow-up criteria for semilinear heat equations on graphs are an active topic, and a treatment of the parabolic p-Laplacian inequality with a time-dependent potential would be a reasonable contribution. The local existence argument and the comparison principle are useful building blocks. However, the central blow-up theorems are not supported: Theorem 1.1 is false as stated (there is a global solution satisfying all stated hypotheses), and the proof of Theorem 1.2 contains a homogeneity error. These issues are load-bearing, so the paper cannot be accepted in its present form.","major_comments":[{"comment":"The proof introduces δ as the minimum of σ on U × [0, T′] for an arbitrary T′ < T, where T is the unknown maximal existence time, and then takes C0 = δ^{-1}. This does not produce a fixed constant: different choices of T′ give different δ, and no uniform-in-time lower bound on σ is assumed. Consequently, the inequalities (3.1)–(3.3) are integrated with a δ that is not fixed on the whole interval of existence. The theorem is false without such a bound. On G = Z with p = 3, take σ(x,t) = e^{-t}, Φ(s) = 10s^2, and u0 ≡ y0 with 0 < y0 < 0.1. Conditions (i)–(iv) hold and (1.4) holds with C0 = 9 and ε0 = 0.5, but the spatially constant solution y(t) = (y0^{-1} - 10(1 - e^{-t}))^{-1} solves (1.3) as an equality and is global. The theorem needs an explicit hypothesis such as inf_{U×[0,∞)} σ ≥ δ > 0, and the proof must use that fixed δ.","section":"§3, Lemma 3.1 / Theorem 1.1"},{"comment":"The construction of the subsolution v = h(t)φ_1(x) is invalid because the p-Laplacian is homogeneous of degree p-1, not degree one. The correct identity is Δ_{p|U}(h(t)φ_1) = h(t)^{p-1}Δ_{p|U}φ_1 = -λ_1 h(t)^{p-1}|φ_1|^{p-2}φ_1, whereas (3.6) uses the linear expression -λ_1 h(t)|φ_1|^{p-2}φ_1. For h ≥ 1 the true diffusion term is more negative than the one used, so the inequality (3.6) does not follow and the Bernoulli ODE (3.5) does not provide a valid subsolution. This invalidates the proof of Theorem 1.2; additionally, the constant δ used in (3.5) is never defined in Lemma 3.2.","section":"§3, Lemma 3.2 / Eqs. (3.5)–(3.6)"},{"comment":"The proof of Theorem 1.2 chooses U = {x0}. For such a set, the Rayleigh quotient in Lemma 2.4 is identically zero because there are no edges inside U, so the argument that λ1 > 0 for connected induced graphs does not apply to the singleton case used in the theorem. The Dirichlet eigenvalue problem on a singleton does have a positive eigenvalue if the boundary edges are included (λ = deg(x0)/μ(x0)), but that value is not obtained by the Rayleigh quotient given in the paper. This is another gap in the proof of Theorem 1.2.","section":"§2, Corollary 2.2 / Theorem 1.2"},{"comment":"The proof restricts the full-graph solution u to U and assumes u(x,t) ≥ 0 on ∂U × [0,T) in order to apply Lemma 2.3 to the auxiliary problem (2.4). Nonnegativity of solutions of the inequality (1.3) for nonnegative initial data is not established anywhere; Lemma 2.1 only gives local existence and continuity in time. Without a maximum principle for the inequality on the whole graph, the application of (1.4) for s ≥ 0 and the boundary comparison are not justified.","section":"§3, proof of Theorem 1.1"}],"minor_comments":[{"comment":"The statement of Lemma 3.2 uses the symbol v both for the solution of (2.4) and for the subsolution h(t)φ_1(x), which makes the statement and proof difficult to follow.","section":"§3, Lemma 3.2"},{"comment":"There is a typo 'on on' in the first sentence, and in the definition of φε the expression 'φ(x)' should read 'φ1(x)'.","section":"§2, Corollary 2.2"},{"comment":"The comparison principle assumes u, v ∈ C^1([0,T)) in time, but the existence lemmas only provide continuity in time; the regularity needed for the blow-up arguments should be stated and proved.","section":"§2, Lemma 2.3"},{"comment":"In the proof of Lemma 2.1, the sentence 'D[v](x,t) ∈ C[0,T] for x ∈ U' should refer to V rather than U, since the lemma concerns the whole vertex set.","section":"§2, Lemma 2.1"}],"recommendation":"reject","confidential_remarks":"The counterexample to Theorem 1.1 is decisive and should be communicated plainly. The homogeneity error in Lemma 3.2 is also fundamental. A revision that adds the missing uniform lower bound on σ and repairs Lemma 3.2 would constitute substantially new mathematics; the comparison principle itself may be salvageable, but the main theorems as stated are not correct."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper has a load-bearing flaw: Theorem 1.1 is false as stated. The stress-test counterexample is valid. On Z with the usual weights, take p=3, σ(x,t)=e^{-t}, Φ(s)=10s^2, and u0 a small positive constant. The p-Laplacian of a constant is zero, so any spatial constant solution satisfies y'=10e^{-t}y^2, whose solution y(t)=(y0^{-1}-10(1-e^{-t}))^{-1} is global for y0<0.1. Hypotheses (i)-(iv) hold, and (1.4) holds with C0=9, ε0=0.5. No finite-time blow-up occurs.\n\nThe source of the problem is in Lemma 3.1: δ is chosen as min σ over U×[0,T'], where T' is an arbitrary time inside the solution's existence interval. For a global solution this does not yield a fixed positive δ, and C0=δ^{-1} cannot be used as a constant determined by σ and u0. The theorem needs an explicit uniform lower bound on σ over U for all time, or over the maximal existence interval with the bound independent of the lifespan.\n\nThat said, the paper is not without merit. The comparison principle for p-Laplacian parabolic inequalities on finite subgraphs (Lemma 2.3) is new, and the proof is essentially correct. The localization strategy — reducing blow-up on the infinite graph to blow-up on a finite subgraph with Dirichlet data — is a real extension of the finite-graph results in Chung-Choi. Lemma 3.1's overall strategy follows [2] closely, but the graph setting and the variable potential make it worth recording once the δ issue is fixed. Theorem 1.2's subsolution construction via the first eigenfunction is standard and, given a uniform lower bound on σ, would work. The same δ-problem infects the definition of h0 in (3.5).\n\nThe gap is central but fixable. I would not accept the paper in its current form. I would send it back for major revision, requiring the authors to state and use a uniform-in-time lower bound on σ over the chosen finite sets, and to verify that all constants (C0, h0) are truly independent of the solution's lifespan. With that change the main results become plausible.\n\nThis paper deserves a serious referee because the comparison principle and the localization framework are useful, and the flaw is an instructive subtlety rather than nonsense. But as it stands, the advertised theorem is false, so the verdict is major revision, not accept.\n\nBest,","headline":"Main theorem is false as stated because the lower bound on σ is not uniform in time, but the comparison principle is sound and the flaw is fixable.","tokens_in":12374,"tokens_out":5578,"would_cite":false,"duration_ms":57023,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R02","35B44","35K55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that on infinite locally finite graphs, any nonzero nonnegative solution of the parabolic p-Laplacian inequality with a source growing at least like $u^{p-1}$ blows up in finite time.","keywords":["parabolic inequality","finite-time blow-up","locally finite graph","p-Laplacian","comparison principle","Dirichlet boundary","first eigenvalue","Bernoulli subsolution"],"falsifier":"Solve, numerically or analytically, the finite-graph Dirichlet problem (2.4) on a path graph with $\\sigma\\equiv 1$, $p=3$, $\\Phi(u)=u^2$ (power $p-1$ with no $\\varepsilon_0$ slack), and constant initial data $u_0\\equiv 1$; if a global solution exists, the strict inequality in Theorem 1.1 is essential, whereas finite-time blow-up would show the threshold is not sharp.","tokens_in":11283,"feed_emoji":"🔥","tokens_out":11998,"duration_ms":118796,"temperature":0.7,"pith_summary":"The paper establishes that nonnegative, nontrivial solutions of the parabolic $p$-Laplacian inequality $u_t - \\Delta_p u \\geq \\sigma(x,t)\\Phi(u)$ on an infinite locally finite graph must blow up in finite time under two sets of conditions: if $\\Phi(s) \\geq (C_0+\\varepsilon_0)s^{p-1}$ with $C_0$ tied to a lower bound of $\\sigma$, or if $\\Phi(s) \\geq C_1 s^q$ with $1<q\\leq p-1$ and the initial data large enough. The proof develops a comparison principle on finite connected subgraphs with Dirichlet boundary data, then pushes blow-up from the subgraph to the whole graph. This transfers classical blow-up theory for parabolic differential inequalities on Riemannian manifolds to the discrete graph setting, with explicit control on the blow-up time.","feed_headline":"Finite-time blow-up proven for graph p-Laplacian inequalities","feed_subtitle":"A subgraph comparison principle proves blow-up for nonzero solutions when the source is superlinear.","key_machinery":"The paper's central tool is the comparison principle (Lemma 2.3) for the parabolic $p$-Laplacian inequality on a finite connected subgraph with zero Dirichlet boundary. It is proved by multiplying the two competing solutions by $e^{-\\lambda t}$, taking a minimum point of the difference, and using the monotonicity of the function $g(a)=|a|^{p-2}a$. This principle lets the authors transplant blow-up from an auxiliary finite-graph problem (2.4) to the original infinite graph. The superlinear blow-up in Lemma 3.1 rests on the maximum function $m(t)=\\max_{x\\in U} u(x,t)$, whose time derivative is controlled by the inequality $m_t \\geq \\delta\\Phi(m)-m^{p-1}$. The subcritical case in Lemma 3.2 uses the first eigenvalue $\\lambda_1$ and positive eigenfunction $\\varphi_1$ of the discrete $p$-Laplacian on $U$, defined via the Rayleigh quotient and a Lagrange multiplier argument (Lemma 2.4), to build an explicit subsolution of Bernoulli type.","core_discovery":"The central claim is Theorem 1.1: if $\\Phi(s) \\geq (C_0+\\varepsilon_0)s^{p-1}$ for $s\\geq 0$, where $C_0$ is the reciprocal of a positive lower bound of $\\sigma$ on a finite connected subgraph and $\\varepsilon_0>0$, then for every nonnegative initial value $u_0$ not identically zero, any solution of (1.3) blows up in finite time at every vertex with $u_0(x_0)>0$. The proof fixes a finite connected subgraph $U$ containing such a vertex, and uses the comparison principle to show that the restriction of $u$ to $U$ dominates the solution $v$ of an auxiliary Dirichlet problem. For $v$, the spatial maximum $m(t)=\\max_{x\\in U} v(x,t)$ is nondecreasing and satisfies $m_t \\geq \\delta\\Phi(m) - m^{p-1} > 0$, which integrates to $t \\leq F(m(0))$ with $F$ finite, forcing blow-up before time $F(m(0))$. Theorem 1.2 covers slower power growth $\\Phi(s) \\geq C_1 s^q$, $1<q\\leq p-1$, by constructing a subsolution $v=h(t)\\varphi_1(x)$ from a Bernoulli ODE whose solution blows up, provided the initial data is at least $h_0$.","pith_inferences":["The borderline case $\\Phi(s)=C_0 s^{p-1}$ with no $\\varepsilon_0$ slack is not covered; whether arbitrary small nonzero data still blow up there would test whether the strict inequality is an artifact of the proof.","The Bernoulli-subsolution construction should extend to coupled systems of $p$-Laplacian inequalities, where the single ODE for $h$ becomes a system and blow-up can be diagnosed by eigenvalue comparisons.","A numerical phase diagram in the $(q, \\|u_0\\|_{\\ell^\\infty})$ plane on an infinite path graph could reveal whether the exponent $p-1$ is the true critical Fujita exponent for locally finite graphs, and whether the large-data threshold $h_0$ is close to sharp.","The comparison principle as stated needs zero Dirichlet boundary; a version with non-zero boundary data would let one localize blow-up to regions where the solution is already large, which is useful for inhomogeneous graphs."],"forward_implications":["Every nonzero nonnegative solution with $\\Phi(s) \\geq (C_0+\\varepsilon_0)s^{p-1}$ explodes at each vertex where the initial value is positive, regardless of how small that value is.","The blow-up time is bounded by $F(u_0)=\\int_{u_0}^{\\infty} \\frac{ds}{\\delta\\Phi(s)-s^{p-1}}$, a finite number that depends only on the vertex maximum of the initial data and the lower bound of $\\sigma$.","For power growth $C_1 s^q$ with $1<q\\leq p-1$, sufficiently large initial data, quantified explicitly by $h_0$, forces blow-up even though the growth is below the $p-1$ threshold.","The comparison principle gives a general recipe: prove a qualitative property on a finite subgraph with Dirichlet boundary, then lift it to any locally finite graph."],"supporting_citations":[{"why":"Supplies the maximum-function integral argument that Lemma 3.1 uses to show the spatial maximum blows up in finite time.","marker":"[2]"},{"why":"The paper states it follows this reference's analytical techniques to implement the comparison principle for the discrete $p$-Laplacian inequality.","marker":"[3]"},{"why":"Provides the Bernoulli equation construction and the blow-up subsolution used in Lemma 3.2 and Theorem 1.2.","marker":"[10]"},{"why":"The Riemannian-manifold parabolic inequality with potential that motivates the graph problem and the growth-rate assumptions on $\\Phi$.","marker":"[14]"},{"why":"Supplies the nodal domain theorem for the graph $p$-Laplacian that ensures the first eigenfunction is sign-definite in Corollary 2.2.","marker":"[17]"},{"why":"The paper adopts its definition of the discrete $p$-Laplacian from this reference.","marker":"[1]"},{"why":"The source of the blow-up definition used in Definition 2.2 and a prior blow-up result for equations on locally finite graphs.","marker":"[12]"}],"fun_headline_variants":["Superlinear source forces p-Laplacian blow-up on graphs","Graph p-Laplacian blows up in finite time for superlinear sources","Blow-up proven for p-Laplacian inequalities on graphs","Finite-time blow-up on graphs for p-Laplacian inequalities","Superlinear source triggers p-Laplacian blow-up on graphs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof needs the potential $\\sigma(x,t)$ to stay above a fixed positive constant $\\delta$ on the chosen finite subgraph for the whole existence time; if $\\sigma$ decays to zero in time, the constant $C_0$ in Theorem 1.1 cannot be fixed and the key inequality $\\delta\\Phi(s)-s^{p-1}>0$ can fail.","fun_headline_variants_meta":{"raw":{"variants":["Superlinear source forces p-Laplacian blow-up on graphs","Graph p-Laplacian blows up in finite time for superlinear sources","Blow-up proven for p-Laplacian inequalities on graphs","Finite-time blow-up on graphs for p-Laplacian inequalities","Superlinear source triggers p-Laplacian blow-up on graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000969,"raw_usage":{"total_tokens":4116,"prompt_tokens":937,"completion_tokens":3179,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":553,"completion_tokens_details":{"reasoning_tokens":3090}},"tokens_in":553,"tokens_out":3179,"duration_ms":23729,"temperature":1.0,"reasoning_tokens":3090,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:51:49.827322+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve, numerically or analytically, the finite-graph Dirichlet problem (2.4) on a path graph with $\\sigma\\equiv 1$, $p=3$, $\\Phi(u)=u^2$ (power $p-1$ with no $\\varepsilon_0$ slack), and constant initial data $u_0\\equiv 1$; if a global solution exists, the strict inequality in Theorem 1.1 is essential, whereas finite-time blow-up would show the threshold is not sharp.","supporting_citations":[{"cited_title":"Chung and M.-J","cited_arxiv_id":null,"evidence_quote":"Supplies the maximum-function integral argument that Lemma 3.1 uses to show the spatial maximum blows up in finite time."},{"cited_title":"Chung and M.-J","cited_arxiv_id":null,"evidence_quote":"The paper states it follows this reference's analytical techniques to implement the comparison principle for the discrete $p$-Laplacian inequality."},{"cited_title":"Kong and M","cited_arxiv_id":null,"evidence_quote":"Provides the Bernoulli equation construction and the blow-up subsolution used in Lemma 3.2 and Theorem 1.2."},{"cited_title":"Mastrolia, D","cited_arxiv_id":null,"evidence_quote":"The Riemannian-manifold parabolic inequality with potential that motivates the graph problem and the growth-rate assumptions on $\\Phi$."},{"cited_title":"Tudisco and M","cited_arxiv_id":null,"evidence_quote":"Supplies the nodal domain theorem for the graph $p$-Laplacian that ensures the first eigenfunction is sign-definite in Corollary 2.2."},{"cited_title":"Amghibech","cited_arxiv_id":null,"evidence_quote":"The paper adopts its definition of the discrete $p$-Laplacian from this reference."},{"cited_title":"Lin and Y","cited_arxiv_id":null,"evidence_quote":"The source of the blow-up definition used in Definition 2.2 and a prior blow-up result for equations on locally finite graphs."}],"review_version":1}