{"id":"36149a84-e4bc-4b79-a1e3-8ab97b2807d8","arxiv_id":"2507.20257","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims pullback attractors for a non-autonomous 2m-th order nonlocal parabolic equation and a positive stationary solution for its autonomous counterpart.","lead":"This paper studies a non-autonomous parabolic equation with a nonlocal 2m-th order Kirchhoff-type diffusion coefficient and claims existence of pullback attractors, a non-autonomous equilibrium, and positive stationary solutions for the autonomous version. It is relevant to researchers working on long-time behavior of nonlocal higher-order diffusion models, though the main attractor proof is incomplete as written.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Comparison hypothesis (13) is not a consequence of (4)-(8): admissible data f=-u, a≡2, f0=-2u, f1=-u, b0=1/2, b1=1 satisfy all stated assumptions but violate (13), so Theorems 2.5 and 2.7 rest on an unverified extra condition.","rationale":"The reader's verdict of REJECT is supported. I focused on (13) because it is the point where the non-autonomous dynamics breaks: global existence via comparison, the invariance of X1+, and the existence of a non-autonomous equilibrium all depend on Theorem 2.5, and Theorem 2.5 depends on (13). The paper does not prove (13), and the admissible example f=-u, a≡2, f0=-2u, f1=-u, b0=1/2, b1=1 satisfies (4)-(8) yet violates (13). This is an internal gap between the stated model assumptions and the comparison hypothesis, not a disagreement with background consensus. I also note that the proof of Theorem 2.7 is circular or at least incomplete: it invokes as a fact that S has a pullback attractor in H^m_0, which is exactly what needs to be established. The autonomous section has separate unresolved points, such as the asserted supersolution C and the energy estimate for E(δφ), but the non-autonomous theorem is the central claim and its proof fails without adding (13) as a separately verified hypothesis. No alternative reading of the text rescues Theorems 2.5-2.7 as written, so the prior verdict should remain REJECT.","tokens_in":10004,"tokens_out":10301,"duration_ms":105276,"concrete_test":"Verify whether (13) follows from (4)-(8) by testing the admissible data f(x,t,u)=-u, a≡2, f0(u)=-2u, f1(u)=-u, b0=1/2, b1=1, with Ω any bounded C^{2m+μ} domain and λ0 chosen so that A+λ0I is positive. Check that all assumptions (4)-(8) hold, then evaluate (13): since g(t,u)=-u/2 and b1f1(u)=-u, the upper inequality reads γu-u/2 ≤ γu-u, which is false for every u>0 and every γ. This settles that (13) is not a consequence of the stated hypotheses, so Theorems 2.5 and 2.7 are not proved for this admissible equation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single most load-bearing point is the comparison assumption (13), introduced in §2.1 as 'From now on, we assume...' and never derived from the hypotheses (4)-(8) on f and a. In (9)-(12) the nonlinearity is g(t,u)=a(∫((A+λ0I)u)u dx,t)^{-1} f(x,φ^{-1}(t),u). Since a may take values in [a0,a1], the factor 1/a can be far from 1, while (13) requires g to be pointwise trapped between b0 f0 and b1 f1 after adding γu. A simple admissible example shows this is not automatic: take f(x,t,u)=-u, a≡2, f0(u)=-2u, f1(u)=-u, b0=1/2, b1=1. Then b0f0=-u, b1f1=-u, so f lies between the endpoint maps; (4)-(8) hold with ρ=1 and suitable constants. But g=-u/2, and the upper inequality in (13) would require γu-u/2 ≤ γu-u for all |u|≤R, which fails for every u>0. Thus the comparison sandwich in Theorem 2.5 is not implied by the stated model assumptions. Since Theorem 2.5 is the only mechanism producing the bounds φ1+≤S(t,s)u0≤φ0+ and the invariance of X1+, the pullback attractor claim in Theorem 2.7 collapses if (13) is not separately enforced. Additionally, the proof of Theorem 2.7 asserts rather than proves the existence of a pullback attractor of S in H^m_0 before restricting to X1+.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a non-autonomous 2m-th order Kirchhoff-type parabolic equation (1)-(2) with a nonlocal diffusion coefficient a and a nonlinearity f. After a time change, the authors reformulate the equation as an abstract semilinear problem (9)-(10) and define an evolution process S. Section 2 introduces comparison hypotheses, proves a sandwich estimate (Theorem 2.5), and claims that S restricted to an order interval X+1 admits a pullback attractor and that a non-autonomous equilibrium exists in C (Theorem 2.7). Section 3 treats the autonomous elliptic problem (16) by variational methods and claims existence of a positive nontrivial weak solution (Theorem 3.2). The central conclusions are not established: the comparison hypothesis (13) is not implied by the standing assumptions and fails for simple admissible data, the proof of Theorem 2.7 invokes as known the very pullback-attractor existence it is supposed to prove, and the variational proof of Theorem 3.2 contains an incomplete supersolution verification and a sign error.","tokens_in":10493,"tokens_out":12158,"duration_ms":119213,"significance":"Higher-order nonlocal parabolic equations and their pullback dynamics are a worthwhile subject, and the comparison-plus-variational strategy is a reasonable route. The paper cites appropriate standard references and attempts to combine order-preserving dynamics with a Kirchhoff-type nonlocal coefficient. However, the main theorems currently rest on an extra comparison hypothesis (13) that is never derived and is violated by simple data satisfying all explicit structural assumptions; Theorem 2.7 is asserted rather than proved; and the energy estimate in Theorem 3.2 contains a sign error. If the comparison assumption were turned into a verified standing condition, and if the pullback-attractor existence were proved directly, the results could be of interest. In the present form, the claims are not supported.","major_comments":[{"comment":"The comparison hypothesis is introduced by 'From now on, we assume...' but it is not derived from (4)-(8). This is load-bearing because g(t,u)=a(∫_Ω((A+λ0I)u)u dx,t)^{-1}f(x,φ^{-1}(t),u), and the factor 1/a changes the sandwich. Explicit data f(x,t,u)=-u, a≡2, f0(u)=-2u, f1(u)=-u, b0=1/2, b1=1 satisfy the stated assumptions (4)-(8) for suitable constants, but then g=-u/2 and (13) would require γu-u≤γu-u/2≤γu-u, which is false for every u>0. Hence Theorem 2.5, and with it the global-existence and invariance claims that depend on it, is not proved under the stated hypotheses.","section":"Section 2.1, Eq. (13)"},{"comment":"The proof says 'The fact that S(·,·) has a pullback attractor in H^m_0(Ω) ensures...' but no argument for the existence of this pullback attractor is given anywhere. That existence is one of the paper's main claims, so the proof is circular. In addition, the final assertion that every global solution in the restricted attractor is a non-autonomous equilibrium in the sense of Definition 2.6 is not justified: no proof is supplied that the zeros of the solution are independent of t or that the solution is non-degenerate as t→±∞.","section":"Section 2.2, proof of Theorem 2.7"},{"comment":"The displayed chain φ+1≤T0(t-s)u0≤S(t,s)u0≤T1(t-s)u0≤φ+0 reverses the order of T0 and T1 relative to Theorem 2.5, which asserts T1(t-s)u0≤S(t,s)u1≤T0(t-s)u2 for u0≤u1≤u2. The following sentence 'since u0 is constant in Ω' does not apply to arbitrary u0∈X+1. Because positive invariance of X+1 is the basis for the restricted pullback-attractor claim, this is a load-bearing gap.","section":"Section 2.2, invariance of X+1"},{"comment":"The proof assumes a bounded supersolution u, but the verification for u≡C is missing; the text 'Here, u≡C>0 is a (weak) supersolution of (16); namely,' breaks off without showing that C satisfies the supersolution inequality. Moreover, the estimate for E(δφ) has a sign error: from F(x,u)≥-c0|u|²+ζ(u) one obtains -∫F≤c0δ²∫|φ|²-∫ζ(δφ), so the coefficient should be a(cδ)μ1+c0 rather than a(cδ)μ1-c0. As written, the negativity of E(δφ) does not follow, so the claimed nontrivial positive solution is not established.","section":"Section 3, Theorem 3.2"},{"comment":"The proof justifies applying the pointwise condition (14) by saying that H^m_0(Ω) is embedded in L∞(Ω). For N>2m, Sobolev embedding gives H^m_0(Ω)⊂L^{2N/(N-2m)}(Ω), not L∞(Ω). Thus the pointwise bound |u|≤R needed to invoke (13) is not available for a general solution in H^m_0(Ω), which further undermines the comparison argument.","section":"Section 2.1, proof of Theorem 2.5"}],"minor_comments":[{"comment":"The right-hand side of (11) is written as (b_i/a_j)f_0, while Definition 2.2 uses only b0/a1 f0; the definitions of the semigroups T0 and T1 are therefore ambiguous.","section":"Section 2.1, Eq. (11) and Definition 2.2"},{"comment":"The function a is first introduced on [0,∞)×R and later on [0,+∞); the notation should be reconciled.","section":"Section 3"},{"comment":"Several displayed formulas suffer from line-break artifacts, including the right-hand side of (11) and the definitions of φ_ε and φ^ε in the proof of Theorem 3.2, which makes the arguments unnecessarily hard to check.","section":"General"},{"comment":"The function ζ is used in the estimate for E(δφ) but is never defined; its dependence on δ should be stated explicitly for the negativity argument to be complete.","section":"Section 3, proof of Theorem 3.2"},{"comment":"The abstract promises 'existence and characterization of pullback attractors', but no characterization result is actually stated or proved; the paper only asserts existence in Theorem 2.7.","section":"Abstract and Introduction"}],"recommendation":"reject","confidential_remarks":"The manuscript appears to be an early draft: the central comparison hypothesis is unverified, Theorem 2.7 is asserted rather than proved, and Section 3 contains an incomplete proof with a sign error. These are not local presentation issues; they affect the main claims. A substantially rewritten version with a verified standing comparison condition, a direct proof of the pullback attractor, and a repaired variational argument would be needed before reconsideration."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Fair warning: this one is a reject for me, but it is not a crank paper. The genuinely new thing is the model class: a 2m-th order non-autonomous Kirchhoff-type nonlocal parabolic equation, and no one seems to have treated pullback attractors for it. The time-change reduction to a semilinear nonlocal problem is standard but sensible, and the functional setting is laid out carefully.\n\nThe problem is that the central comparison result (Theorem 2.5) rests on hypothesis (13), which is introduced as \"we assume...\" and never derived from the model hypotheses (4)-(8). That is not a harmless gap: the stress-test counterexample is right. Take f(x,t,u)=-u, a≡2, f0=-2u, f1=-u, b0=1/2, b1=1. All of (4)-(8) hold, b0f0≤f≤b1f1 holds, but g(t,u)=f/a=-u/2, and the upper inequality in (13) fails for every u>0. So the sandwich estimate is an extra condition, not a consequence. Since Theorem 2.5 is the only thing producing the X_1^+ invariance, Theorem 2.7 collapses unless (13) is enforced. And Theorem 2.7's proof is circular: it invokes \"the fact that S(·,·) has a pullback attractor in H^m_0\" without proving it anywhere. There are also smaller errors: in the invariance display the T0/T1 order is reversed relative to Theorem 2.5, and the claim that H^m_0 embeds in L∞ is false when N>2m (which is exactly their standing assumption).\n\nSection 3 is weaker. The supersolution for (16) is announced (\"Here, u C≡C>0 is a (weak) supersolution; namely,\") but never actually verified. The energy estimates have gaps, and the condition a(0)μ1