{"id":"c5041e03-86e0-409e-b050-2471666c4a4a","arxiv_id":"2608.07970","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Unknown initial data, linear potentials and analytic nonlinearities in nonlinear Kirchhoff plate equations are uniquely and stably recoverable from active or passive boundary measurements.","lead":"This mathematics paper proves that unknown initial data, time-dependent coefficients, and nonlinear terms in Kirchhoff plate equations can be recovered from boundary measurements. It matters because plate equations model thin elastic structures, and no prior work recovered these multiple unknowns simultaneously for this fourth-order time-domain system.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.3 Step 1 uses a false support claim: q̃1−q̃2 is not supported in Ω×[t1,t2], because outside that slab both terms equal ∂_z g0(x,t,w_j) and w1≠w2 has not yet been ruled out.","rationale":"I read the paper in good faith and agree that it is a substantial contribution with many correct-looking components: the forward well-posedness section is detailed, the linear coefficient recovery in Theorem 1.2 follows a standard GO/Runge pattern, and the observability inequality of Lemma 4.1 is plausibly exactly the result imported from [ZZ06]. The reader's weakest assumption is about the unproved import; that is a legitimate verification request, but it is not the place where the argument actually breaks internally. The break is Step 1 of Theorem 1.3: the asserted support of q̃1−q̃2 is false, because evaluating ∂_z g_j at two different background solutions w1 and w2 outside the time slab does not cancel. This support claim is load-bearing: it restricts the crucial integral to (t1,t2), where the Runge approximation and GO solutions are available, and it underlies the conclusion q̃1=q̃2 in Ω_T that is fed into every higher-order linearization step. Since w1=w2 is only established later, the proof is circular at this point. The flaw is localized and might be repairable by proving derivative equality only on [t1,t2] and controlling the global potential G in Step 4 through the analyticity of g0 outside the slab. That is why I recommend UNVERDICTED rather than REJECT: the claimed theorem is not established by the manuscript as written, but a controlled revision may well repair it. My concrete test settles that the support claim itself is false; the remaining question is whether the surrounding argument can be restructured without it.","tokens_in":54290,"tokens_out":18881,"duration_ms":205627,"concrete_test":"Test the support claim directly. Take n=1, Ω=(0,1), choose an analytic function g0 with ∂_z^2 g0 not identically zero, for instance g0(z)=z^2, and set g1=g2=g0, so both belong to G^{t1,t2}_{g0} for any t1<t2. Pick two distinct small initial data (η_11,η_21) and (η_12,η_22) and let w1,w2 be the corresponding solutions of (1.7) with zero boundary data, which exist by Theorem 2.5. For t outside [t1,t2], q̃1−q̃2 = ∂_z g0(w1)−∂_z g0(w2) = 2(w1−w2); energy and uniqueness arguments show this is generically nonzero on a time interval of positive length. This directly contradicts the asserted support property. To decide whether the theorem is salvageable, also check whether the Fourier/Runge step in Step 1 can be rewritten using the genuinely supported quantity ∂_z(g1−g2)(x,t,·) in place of q̃1−q̃2; if not, the proof of Theorem 1.3 lacks a valid basis.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"In the proof of Theorem 1.3 (Section 4.4, Step 1), the paper sets q̃_j = ∂_z g_j(x,t,w_j), with w_j the ε=0 solution associated to initial data (η_1j,η_2j). It then asserts that g1,g2 ∈ G^{t1,t2}_{g0} implies supp(q̃1−q̃2) ⊂ Ω×[t1,t2]. This implication is false. By definition (1.15), for t outside [t1,t2] one has g1(x,t,z) = g2(x,t,z) = g0(x,t,z) for every z, so q̃1−q̃2 = ∂_z g0(x,t,w1)−∂_z g0(x,t,w2), which need not vanish unless w1=w2. The equality w1=w2 is not known until the end of Step 4. The subsequent reduction of the identity ∫_0^T (q̃1−q̃2) v_1^(k) y dxdt to an integral over (t1,t2), and the GO/Fourier argument that concludes q̃1=q̃2 in Ω_T, both rely on this support assertion. The same unsupported reduction is needed in Steps 2 and 3 for ∂_z^M g1(x,t,w1)−∂_z^M g2(x,t,w2). Thus the central simultaneous recovery theorem for nonlinearities and initial data is not proved as written. I do not see a comparable internal obstruction in the import of Lemma 4.1; the concrete problem here is inside the nonlinear argument itself.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies forward and inverse boundary value problems for nonlinear Kirchhoff plate equations of the form (I−γΔ)∂t²u + Δ²u = g(x,t,u) under hinged boundary conditions. The contributions are: (i) global well-posedness for semilinear equations with a wide class of nonlinearities satisfying Carathéodory and subcritical growth conditions; (ii) local well-posedness for analytic nonlinearities in dimensions n=1,2,3; (iii) stable recovery of initial data from a single passive boundary measurement when the nonlinearity is known; (iv) simultaneous recovery of a time-dependent, possibly unbounded potential and initial data from infinitely many active boundary measurements; and (v) simultaneous recovery of an analytic nonlinearity and initial data, using geometric optics solutions, a Runge approximation, and higher-order linearization around nonzero solutions. The main theorems are Theorem 1.1 (passive stable recovery), Theorem 1.2 (linear coefficient and initial data recovery), and Theorem 1.3 (nonlinearity and initial data recovery). The paper also contains counterexamples showing that passive measurements cannot recover initial data when the source or coefficient is unknown.","tokens_in":54592,"tokens_out":14999,"duration_ms":170111,"significance":"If the proofs are completed, the results represent a substantial advance in inverse problems for time-domain plate equations. The well-posedness theory is developed in detail with explicit energy and semigroup arguments, and the geometric optics construction includes careful remainder estimates (Lemma 3.1). The simultaneous recovery of unbounded time-dependent potentials and initial data, and of nonlinearities and initial data, goes beyond the existing literature on Kirchhoff and Euler–Bernoulli plates. The paper also provides useful counterexamples for passive measurements and a Runge approximation result (Proposition 4.2) that is of independent interest. However, the proof of Theorem 1.3 contains a false support assertion and an invalid Taylor expansion at load-bearing points; these must be repaired before the central simultaneous-recovery claim can be considered established.","major_comments":[{"comment":"The claim that g1,g2 ∈ G^{t1,t2}_{g0} implies supp(∂_z g1(·,·,w1)−∂_z g2(·,·,w2)) ⊂ Ω×[t1,t2] is false. By definition (1.15), outside the slab [t1,t2] both g1 and g2 equal g0 as functions of z, so the difference equals ∂_z g0(·,·,w1)−∂_z g0(·,·,w2), which need not vanish before w1=w2 is proved. The subsequent reduction of ∫_0^T (q̃1−q̃2)v_1^{(k)}y dxdt to an integral over (t1,t2), and the GO/Fourier conclusion q̃1=q̃2 in Ω_T, depend on this support assertion. The same unsupported reduction is used in Steps 2 and 3 for ∂_z^M g1(x,t,w1)−∂_z^M g2(x,t,w2), so the central simultaneous recovery theorem is not proved as written.","section":"Section 4.4, Step 1 (proof of Theorem 1.3)"},{"comment":"The Taylor identity displayed in (4.23) is invalid. For analytic g1,g2, the difference g1(w1)−g2(w2) is not equal to Σ_{k≥1} (−1)^k/k! [∂_z^k g2(w2) w2^k − ∂_z^k g1(w1) w1^k]; the right-hand side mixes Taylor coefficients at 0 with evaluations at w1,w2. The subsequent bound on G(x,t) and the observability step for w̃=w1−w2 are therefore unsupported. A repair is available: from ∂_z^k g1(x,t,w1)=∂_z^k g2(x,t,w2) for all k, the entire function z↦g1(x,t,z)−g2(x,t,z+w2−w1) has all derivatives vanishing at z=w1, hence is identically zero, so g1(x,t,w1)=g2(x,t,w2) and w̃ satisfies a homogeneous equation. This route needs to be written out explicitly.","section":"Section 4.4, Step 4, Eq. (4.23)"},{"comment":"The induction step that turns (4.22) into ∂_z^M g1(x,t,w1)=∂_z^M g2(x,t,w2) is sketched rather than proved. It is not specified how the Runge approximations and GO solutions for v^{(2)},...,v^{(M)} and y are chosen for even and odd M, how the product of the leading-order amplitudes is handled, or how the Riemann–Lebesgue/analytic-continuation argument is completed for the resulting integral identity. Since this induction is the only mechanism producing the all-derivative identities used in Step 4, the argument needs to be supplied in full.","section":"Section 4.4, Step 3"},{"comment":"The observability inequality is imported from [ZZ06, Theorem 3] with an explicit constant exp(C∥b∥_{L∞(0,T;L^p)}^{2p/(6p−5n)}) and with the geometry rescaled by γ, but the paper does not verify that the hypotheses of [ZZ06] are satisfied after the metric change g=γ ds² and conditions (1.3)–(1.4). Because Lemma 4.1 underpins Theorem 1.1, Proposition 4.2, and the final steps of Theorem 1.3, the authors should either prove the inequality or provide a precise statement of how it follows from the cited theorem.","section":"Section 4.1, Lemma 4.1"}],"minor_comments":[{"comment":"The estimate contains the typo '∥w0∥H2(Ω)∥+∥w1∥H1(Ω)' with an extra norm symbol.","section":"Section 2.2, Step 3"},{"comment":"The term ∥∂_t u(t)∥²_{L²(Ω)} is redundant in the sum with ∥∂_t u(t)∥²_{H¹(Ω)}; harmless but should be cleaned up.","section":"Section 2.2, Eq. (2.24)"},{"comment":"The notation for the linearized solutions is confusing: v_j^{(k)} is used both for the solution of the linearized equation with boundary data h_{1k},h_{2k} and later for its difference with the other index; the two objects should be denoted differently.","section":"Section 4.4, Step 1"},{"comment":"The cut-off construction is presented as an alternative for the linear coefficient recovery; the abstract's phrasing about 'addressing the scenario of vanishing initial data' could mislead readers into expecting it to cover the nonlinear recovery in Theorem 1.3, so the scope should be stated explicitly.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The support-claim counterexample raised in review is real and should be addressed first. If the authors repair Step 1 by first recovering the initial data on the exterior intervals [0,t1] and [t2,T] via the observability inequality, and replace (4.23) by the shift-of-argument argument, the overall strategy appears plausible. The induction in Step 3 and the verification of Lemma 4.1 also need to be written out before I can certify Theorem 1.3."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a serious paper on inverse boundary value problems for nonlinear Kirchhoff plates, with real new material—global well-posedness, GO solutions for the fourth-order eikonal equation, and a Runge approximation in a smaller space. Theorems 1.1 and 1.2 look credible. But the proof of Theorem 1.3, the simultaneous recovery of nonlinearity and initial data, has a false support claim in Step 1, and that is load-bearing.\n\nThe claim: because g1=g2=g0 outside [t1,t2], the difference ∂z g1(x,t,w1)−∂z g2(x,t,w2) is supported in Ω×[t1,t2]. That is not true. Outside the slab, the difference equals ∂z g0(x,t,w1)−∂z g0(x,t,w2), and vanishing requires w1=w2, which is exactly what the theorem is trying to prove. The reduction of the first-order integral to (t1,t2), and the subsequent GO/Fourier step, depend on that support assertion. The same issue recurs in the higher-order steps. I don't see a way around it in the text as written; the parity discussion for even M ('add a GO solution') is also too terse to verify. So the main theorem is not proved as written.\n\nOther soft spots: Lemma 4.1 imports an observability inequality from [ZZ06] with an explicit L^p range and exponent, but the paper doesn't quote the exact theorem or check the geometric conditions (1.3)-(1.4) match [ZZ06]'s hypotheses. That's a verification burden, not necessarily a flaw. The compatibility condition in (1.8) writes η2|Γ = h2(0), which is inconsistent with (2.14), where η2|Γ = ∂t h1(0); likely a typo but should be fixed. The abstract says 'readily generalizable' while Section 5 says 'We believe'; soften the abstract.\n\nWhat the paper does well: the GO construction for (I−γΔ)∂t^2+Δ^2 with eikonal |∇φ|^4−γφ_t^2|∇φ|^2=0 is new and carefully done; Lemma 3.1's remainder estimates look right. The forward well-posedness section is detailed and mostly self-contained. Theorem 1.1's stable recovery from one passive boundary measurement is a solid result. The Runge approximation in a smaller space is valuable independent of the inverse theorems.\n\nBottom line: this deserves a serious referee—there is enough original machinery here to warrant the time—but only after the authors fix the support argument in Theorem 1.3 or restructure the proof to avoid it. If they can't, the simultaneous-recovery theorem falls, and the paper would have to be recast around Theorems 1.1 and 1.2. Send it back for a major revision.","headline":"Substantial new GO/Runge machinery for plate inverse problems, but Theorem 1.3 has a false support claim that leaves the headline recovery theorem unproved as written.","tokens_in":55171,"tokens_out":3627,"would_cite":false,"duration_ms":40634,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","26A33","42B37"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that passive boundary data stably recover initial data for nonlinear Kirchhoff plates, and that active boundary data determine unbounded time-dependent potentials, or analytic nonlinearities together with initial data.","keywords":["inverse problems","Kirchhoff plate equation","Runge approximation","geometric optics solutions","higher-order linearization","unbounded time-dependent potentials","initial data recovery"],"falsifier":"A concrete check: verify whether [ZZ06, Theorem 3] holds verbatim for potentials $b\\in L^\\infty(0,T;L^p(\\Omega))$ with $p\\in[5n/2,\\infty]$, with the constant $\\exp(C\\|b\\|^{2p/(6p-5n)})$, on every domain satisfying (1.3)-(1.4). A counterexample—two initial states producing indistinguishable boundary traces for some such $b$ and $T>T_0$—would overturn Lemma 4.1 and hence Theorems 1.1, 1.2(2), and 1.3; a numerical verification of the inequality on a simple domain would support them.","tokens_in":54046,"feed_emoji":"🧩","tokens_out":11083,"duration_ms":112513,"temperature":0.7,"pith_summary":"Boundary measurements of a vibrating Kirchhoff plate carry enough information to reconstruct what happens inside the plate. The paper proves that with a known nonlinearity $f$, one passive boundary record stably recovers the initial displacement and velocity; with many active boundary inputs, an unbounded time-dependent potential $q$ can be determined, and in the nonlinear case an analytic nonlinearity $g$ together with the initial state can be recovered simultaneously. It also establishes global well-posedness for the semilinear forward problem, a prerequisite for any of these inverse statements. Why this matters: plate models are standard in structural mechanics, and these results say the internal forces and initial deformation can be calibrated from edge measurements alone, without a priori knowledge of the interior. The proof transfers higher-order linearization, geometric-optics solutions, and Runge approximation from second-order wave equations to fourth-order hinged plates with rotational inertia $\\gamma\\Delta\\partial_t^2$.","feed_headline":"One boundary map recovers plate nonlinearity and initial data","feed_subtitle":"Edge data determine what forces and motion hide inside an unknown plate.","key_machinery":"The main engine is a package rather than a single identity. Geometric optics solutions $v=e^{i\\sigma\\varphi}a+r_\\sigma$, built from plane-wave phases $\\varphi=x\\cdot\\theta+t$ solving the eikonal equation $|\\nabla\\varphi|^4-\\gamma\\varphi_t^2|\\nabla\\varphi|^2=0$ and from amplitudes along transport equations, encode coefficient information in Fourier integrals that vanish only if the coefficients agree. A Runge approximation (Proposition 4.2), derived from the observability inequality of [ZZ06] imported as Lemma 4.1, lets the authors approximate arbitrary plate solutions by solutions that vanish at $t=0$ or $t=T$, exactly the regime where GO test functions would otherwise be unusable. Higher-order linearization around nonzero background solutions $w_j$ extracts $\\partial_z^k g_j$ one derivative at a time from the boundary map, and the observability inequality finally forces the background solutions to coincide, completing the recovery of $g$ on $\\Omega_T\\times\\mathbb{C}$. In Section 5 a cut-off procedure supplies explicit GO solutions with zero initial/terminal data as an alternative to Runge approximation.","core_discovery":"Stated in the paper's own terms, the central discovery is threefold. Theorem 1.1: for $f$ in the admissible class with growth exponent $r_0$, the passive map $(\\partial_\\nu u,\\partial_\\nu\\Delta u)|_{\\Gamma_{0T}}$ controls the initial pair in $H^3(\\Omega)\\times H^2(\\Omega)$ via estimate (1.18), so initial data alone are stably recoverable. Theorem 1.2: for the linear equation $(I-\\gamma\\Delta)\\partial_t^2 u+\\Delta^2 u+q u=0$ with $q\\in L^\\infty(0,T;L^{5n/2}(\\Omega))$ spatially unbounded, the active maps $\\Lambda_{\\Gamma,T}$ and $\\tilde\\Lambda_{\\Gamma,T}$ determine $q$; when $q$ coincides with a known $q_0$ outside $(t_1,t_2)$, these maps simultaneously determine $q$ and the initial data. Theorem 1.3: for $n\\in\\{1,2,3\\}$ and analytic nonlinearities $g_1,g_2$ in the admissible set $G^{t_1,t_2}_{g_0}$, agreement of the active maps on all small boundary data forces $g_1=g_2$ on $\\Omega_T\\times\\mathbb{C}$ and equality of both initial pairs, and the same holds for the DN-type map without $u|_{t=T}$. The paper also gives counterexamples showing why passive measurements cannot recover initial data when the nonlinearity or the coefficient is unknown.","pith_inferences":["Editorially, the same inductive scheme should recover the first $M$ Taylor coefficients of $g$ when analyticity is weakened to $C^M$ regularity, with only the higher-order Taylor tail left undetermined; analyticity is what buys the full $\\Omega_T\\times\\mathbb{C}$ identification.","The Fourier-transform step in the GO construction suggests the support condition $\\operatorname{supp}(g_1-g_2)\\subset\\Omega\\times(t_1,t_2)$ is what forces the active-data regime; without it, the identity (4.11) would not reduce to a localized Fourier transform.","The cut-off construction of Section 5 shows that explicit zero-initial-data GO solutions exist whenever the coefficient is known near $t=0$ or $t=T$, so the Runge approximation can be bypassed in models without an observability inequality.","A quantitative version of Lemma 4.1 with explicit geometric dependence would convert these uniqueness results into stability estimates for the nonlinear problem and would indicate how many boundary measurements are needed in practice."],"forward_implications":["Initial data for semilinear Kirchhoff plates can be recovered stably from a single passive boundary record whenever the nonlinearity is known.","Unbounded, time-dependent potentials are recoverable from active boundary measurements, removing an earlier boundedness restriction for plate inverse problems.","For $n=1,2,3$, analytic nonlinearities and initial states are simultaneously determined by infinitely many active boundary measurements, including the variant measuring only $(\\partial_\\nu u,\\partial_\\nu\\Delta u)$ on the side boundary.","The passive-data counterexamples imply that absence of active inputs creates genuine non-uniqueness, so the simultaneous results inherently require active measurements.","The methods are stated to extend to other boundary conditions and to Euler-Bernoulli plates with unbounded potentials."],"supporting_citations":[{"why":"Supplies the observability inequality for linear Kirchhoff plates with potentials; Lemma 4.1 and the constants $T_0,T$ in (1.3)-(1.4) are imported from it, so every recovery result depends on it.","marker":"[ZZ06]"},{"why":"Gives the higher-order linearization around nonzero solutions and Runge approximation scheme for semilinear wave equations that the nonlinear plate proof adapts; it is the template for simultaneous recovery of nonlinearity and sources.","marker":"[LLL24]"},{"why":"Supplies the sharp trace result $\\partial_\\nu\\partial_t u\\in L^2(\\Gamma_T)$ used to make the passive and active boundary measurements well-defined.","marker":"[LT99]"},{"why":"Foundational for the semigroup and fractional-power treatment of the Kirchhoff operator; Theorem 2.2's well-posedness is built on this abstract formulation.","marker":"[LT91]"},{"why":"Used in the abstract semigroup analysis of the plate operator, including the strong $C_0$-semigroup and domain characterizations.","marker":"[AL98]"},{"why":"Provides fractional interpolation identifications for $\\Delta^2$, giving the $H^2$ and $H^3$ norm equivalences that run through the energy estimates.","marker":"[Gri67]"},{"why":"Supplies the Green-map and trace extension machinery that turns nonhomogeneous hinged boundary data into compatible interior data.","marker":"[LM12]"}],"fun_headline_variants":["Active edge data recover plate nonlinearity and initial data","Infinite edge probes unlock plate's hidden forces and motion","Many boundary measurements determine plate unknowns","Simultaneous recovery of plate force and motion from edge data","Active measurements reveal both plate forces and initial state"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole edifice rests on Lemma 4.1, an observability inequality imported from [ZZ06] asserting that, for linear hinged Kirchhoff plates with potential $b\\in L^\\infty(0,T;L^p(\\Omega))$, $p\\in[5n/2,\\infty]$, and $T>T_0$, the boundary traces control the initial state with an explicit exponential factor; if that inequality fails in this exact range or for the geometric constants (1.3)-(1.4), the stability estimate, the Runge approximation, and every recovery theorem collapse with it.","fun_headline_variants_meta":{"raw":{"variants":["Active edge data recover plate nonlinearity and initial data","Infinite edge probes unlock plate's hidden forces and motion","Many boundary measurements determine plate unknowns","Simultaneous recovery of plate force and motion from edge data","Active measurements reveal both plate forces and initial state"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000566,"raw_usage":{"total_tokens":2740,"prompt_tokens":1062,"completion_tokens":1678,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":678,"completion_tokens_details":{"reasoning_tokens":1604}},"tokens_in":678,"tokens_out":1678,"duration_ms":14185,"temperature":1.0,"reasoning_tokens":1604,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:38:47.148574+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check: verify whether [ZZ06, Theorem 3] holds verbatim for potentials $b\\in L^\\infty(0,T;L^p(\\Omega))$ with $p\\in[5n/2,\\infty]$, with the constant $\\exp(C\\|b\\|^{2p/(6p-5n)})$, on every domain satisfying (1.3)-(1.4). A counterexample—two initial states producing indistinguishable boundary traces for some such $b$ and $T>T_0$—would overturn Lemma 4.1 and hence Theorems 1.1, 1.2(2), and 1.3; a numerical verification of the inequality on a simple domain would support them.","supporting_citations":[],"review_version":1}