{"id":"a5768c4a-c841-480e-aa94-fdaf6886535d","arxiv_id":"1908.03729","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For linear evolution equations on time-dependent intervals, the unknown Neumann boundary values are characterized as the unique solution of a system of Volterra integral equations with explicit kernels.","lead":"This paper derives a system of integral equations that computes the unknown boundary derivatives of solutions to the heat and Schrödinger equations when the solution domain is a moving interval. If correct, it provides a complete Dirichlet-to-Neumann map for time-dependent boundaries, a key ingredient for solving and simulating moving-boundary problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2's ε→0 limit is proven for kernels only; convergence of the regularized solutions (f^ε,g^ε) is asserted without compactness or uniform bounds, leaving the Schrödinger characterization unsupported.","rationale":"The paper contains a useful formal derivation and the heat-equation part is on solid classical ground. My reading agrees with the reader that the central weak point is the ε→0 passage in Theorem 2. I do not see an internal inconsistency, and the result may well be true; the issue is that the proof as written does not establish the existence of the limiting solution. The scalar reduction is not enough because the sequence of solutions is not fixed. The growth of the L^1 norm of the off-diagonal kernels indicates that a naive uniform bound is nontrivial. I would keep the verdict CONDITIONAL (i.e., no change from the reader): the paper should be accepted only after the authors supply a complete convergence argument (or an alternative proof of well-posedness of (4.4)-(4.5) directly).","tokens_in":14733,"tokens_out":10922,"duration_ms":110730,"concrete_test":"Prove or disprove existence of the limit by establishing an ε-uniform bound for (4.1) under (1.14). Specifically, derive a Gronwall-type estimate for ||(f^ε,g^ε)||_{C[0,T]} with a constant independent of ε (or, if the raw kernel norm diverges, use the integration-by-parts representation to obtain uniform L^2 or Hölder bounds), then extract a convergent subsequence and verify it satisfies (4.4)-(4.5). A direct test: compute the resolvent kernel of (4.1) for the linear-boundary case (4.6) and show sup_ε ||·|| < ∞; if the bound fails, the asserted limit may not exist.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem 2 (§4.2), the proof of the ε→0 limit is incomplete at the level of the solution, not just the kernels. For each ε>0, (4.1) has a unique solution (f^ε,g^ε) by classical Volterra theory. The paper then proves, for a fixed smooth h, that ∫ K12(t,s,ε)h(s)ds converges by dominated convergence to an expression containing h and h′ (and similarly for K21). It concludes: 'Hence for f1(t)=lim f^ε(t), g1(t)=lim g^ε(t), using (4.1) we find (4.4)-(4.5).' No argument establishes existence of these limits. The vector extension is not immediate: the two components are coupled, and the limiting equations contain f′ and g′, so even uniform convergence would not pass to the derivative terms without a Hölder or C^1 bound. No compactness, equicontinuity, or ε-uniform Gronwall estimate is supplied. This is not a cosmetic gap: for j≠m, |K_jm(t,s,ε)| has L^1 norm growing like ε^{-1/4} near s=t (scale u=√ε w), so the resolvent bound from (4.1) alone diverges as ε→0. Thus the characterization of the Neumann data as the unique solution of (4.4)-(4.5) is not proven. Condition (1.14) only ensures H1,H2≠0, making the limiting kernels well defined; it supplies no convergence of the regularized solutions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a Fokas-type (unified transform) analysis for linear evolution PDEs on a time-dependent interval l1(t)<x<l2(t), 0<t<T. The authors derive a global relation from a divergence form and invert the Fourier transform to obtain formal integral representations of the unknown Neumann boundary values. For the heat equation they obtain a system of two linear Volterra integral equations, (1.7), with explicitly computed Gaussian kernels, and prove in Theorem 1 that this system has a unique C^1 solution. For the linear Schrödinger equation they obtain a system (1.10) containing an ε-regularized kernel; Theorem 2 asserts that, under the convexity condition (1.14) or the linear-boundary condition (4.6), the ε-limit exists and yields a generalized Volterra system (4.4)-(4.5) with a unique solution. The paper also sketches the reduction of the general evolution PDE case to the global relation and cites prior work for uniqueness of the resulting initial-boundary value problem.","tokens_in":15000,"tokens_out":3032,"duration_ms":31865,"significance":"If the main results are correct, the paper gives an explicit characterization of the Dirichlet-to-Numann map for the heat and linear Schrödinger equations on time-dependent intervals, extending earlier half-line results of Fokas-Pelloni and Xia. The heat-equation part appears well supported: the global-relation derivation is self-contained and the kernel computations in Claims 1-2 are standard, with the weakly singular Volterra theory invoked appropriately. The claimed Schrödinger result is more delicate because the limiting kernels are obtained through an ε-regularization and the limiting equations contain first derivatives of the unknown functions. The paper also honestly delineates the restrictive class of boundary curves for which the LS result is claimed. Because the proof of Theorem 2 has a genuine gap in the ε-limit passage, the significance of the Schrödinger characterization rests on an unproven step; the heat-equation contribution stands independently.","major_comments":[{"comment":"The proof of Theorem 2 does not establish the existence of the limits f1(t)=lim f^ε_1(t) and g1(t)=lim g^ε_1(t) that are used to pass from the regularized system (4.1) to the limiting system (4.4)-(4.5). The text proves convergence of the kernels for a fixed smooth h by dominated convergence, but no argument is given for convergence of the solutions of the coupled Volterra system; the statement that the vector extension is 'immediate' is not supported. This is load-bearing, since the characterization of the unknown Neumann values for the Schrödinger equation is exactly the unique solvability of (4.4)-(4.5).","section":"§4.2, Eq. (4.1)-(4.5)"},{"comment":"Even if one had uniform convergence of f^ε and g^ε, the limiting equations (4.4)-(4.5) contain f'_1 and g'_1 in the integrands, so uniform convergence alone would not justify passing the ε-limit through the derivative terms. A Hölder or C^1 bound, or an alternative compactness argument, is required. The manuscript does not supply such a bound, and the scalar-case reduction cannot address the coupling between the two components.","section":"§4.2, scalar reduction and derivative terms"},{"comment":"The claim that (4.4)-(4.5) is a 'generalised Volterra integral equation of the second kind with a weakly integral kernel' and that a modification of the iterative proof works is only supported by a reference to Brunner. The hypotheses of the relevant existence theorem are not verified: the equations are not in standard Volterra form because of the derivative terms, and the kernels contain H1(t,s)^{-1} and H2(t,s)^{-1}, whose regularity is only shown under (1.14) or (4.6). The proof should either state and verify a precise theorem from Brunner or provide a self-contained fixed-point argument.","section":"§4.2, existence/uniqueness of the generalized Volterra system"}],"minor_comments":[{"comment":"There is a typo: 'Assume the the boundary functions' should read 'Assume the boundary functions'.","section":"Theorem 2 statement"},{"comment":"In the inequality proving H1(t,s)<0, the term 'l′(s)' should be 'l′_2(s)' for consistency with the definition (4.3).","section":"§4.2, H1 estimate"},{"comment":"In the definition of K_jj, the symbol 'ε > 0' appears, but K_jj does not depend on ε; the condition should probably be '0<s<t<T' only.","section":"Equation (1.12)"},{"comment":"The phrase 'Theorem (2)' should be 'Theorem 2'.","section":"Remark 2"},{"comment":"The notation E12(t,s,ε) is introduced for the exponential, but the subsequent display uses E12(t,s) without ε in some places; the dependence on ε should be tracked consistently.","section":"Section 4.2, text near (4.2)"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper extends the Fokas unified transform to linear evolution PDEs on a time-dependent interval with two moving boundaries. The heat equation part is essentially complete: the unknown Neumann data are characterized as the unique solution of a system of two Volterra integral equations with explicit, weakly singular kernels, and a classical existence theorem applies. That is a genuine step beyond the half-line analysis of Fokas–Pelloni (2012). The Schrödinger case is where I have real reservations.\n\nThe formal derivation of the integral equations from the global relation is coherent, and Claims 1–4 contain the honest Gaussian computations. The regularization of the strongly singular kernels via ε is a sensible device, and the limiting equations (4.4)–(4.5) are formally right under the geometric condition (1.14) or the linear-boundary condition (4.6). The citation pattern is reasonable: uniqueness is delegated to Xia (2019), and the Volterra machinery to Miller–Feldstein, which is acceptable for a sketch.\n\nThe soft spot is Theorem 2. The proof shows convergence of the kernels as ε → 0, but the existence of the limits of the solutions (f^ε, g^ε) is never established. The sentence 'Hence for f1(t)=lim f^ε(t), g1(t)=lim g^ε(t) ...' is exactly where the argument jumps. The stress-test note is right: the vector extension is not immediate. The limiting equations contain f′ and g′, so even uniform convergence of f^ε would not pass to the derivative terms without a Hölder or C^1 bound. The ε-regularized system involves kernels with L^1 norms that may grow like ε^{-1/4} near the diagonal, so a naive resolvent bound diverges. You need a compactness/equicontinuity argument, or an ε-uniform Gronwall estimate, to justify the limits. This is not a cosmetic gap; it is a missing step in the proof of the main Schrödinger result.\n\nI should also note that Theorem 1's proof is a reference to earlier work, not self-contained, but that is standard and acceptable. The heat equation part is solid.\n\nWho is this for? People working on the Fokas method and moving-boundary problems. They will find the heat equation result useful and the Schrödinger formulation a promising starting point, but they should treat Theorem 2 as conditional pending a complete proof.\n\nI would send it to peer review. A serious referee could help the authors fill the gap or see that the claim is actually false. The paper deserves engagement, not a desk reject. I would cite the heat equation result, but not the Schrödinger theorem as proven until the ε→0 limit is sorted out.","headline":"A competent extension of the unified transform to moving finite intervals, with a solid heat equation result and a Schrödinger theorem whose proof skips the hard convergence step.","tokens_in":15574,"tokens_out":2770,"would_cite":true,"duration_ms":27044,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K05","35Q41","35R37","45D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the missing boundary data for evolution equations on moving intervals can be recovered from a two-by-two system of integral equations.","keywords":["linear evolution equations","time-dependent interval","Dirichlet-to-Neumann map","heat equation","linear Schrödinger equation","Volterra integral equations","global relation","moving boundary"],"falsifier":"Take a boundary pair satisfying (1.14), for example $l_1(t)=-t+t^2/2$ and $l_2(t)=1+t-t^2/2$ on a small interval, solve the regularized system (4.1) numerically for decreasing $\\varepsilon>0$, and check whether the solutions converge to a limit satisfying (4.4)–(4.5). If the limit fails to exist or violates the equations, the Schrödinger theorem’s proof has a hole; for the heat equation, testing (1.7) against an exact solution with known boundary fluxes would confirm or falsify the uniqueness claim.","tokens_in":1728,"feed_emoji":"📏","tokens_out":2063,"duration_ms":100079,"temperature":0.7,"pith_summary":"This paper claims that for linear evolution equations on a finite interval whose endpoints move in time, the missing boundary data—the spatial derivatives of the solution at the two moving endpoints—are determined uniquely by the prescribed initial and Dirichlet data. For the heat equation the claim holds for arbitrary differentiable, non-crossing boundary curves; the unknown Neumann values solve a system of two linear Volterra integral equations with explicit kernels. For the linear Schrödinger equation the same characterization holds only when the boundary curves satisfy a convexity condition or are linear with compatible slopes. If true, this completes the Dirichlet-to-Neumann construction for these problems and yields an explicit solution representation on the moving domain. Moving-boundary problems arise in melting, diffusion, and interfacial dynamics, where the evolution of boundary fluxes is the central unknown.","feed_headline":"A 2x2 integral system recovers boundary fluxes on moving domains","feed_subtitle":"Heat and Schrödinger boundary slopes on moving intervals are uniquely determined by such a system.","key_machinery":"The key machinery is the global relation (2.8), a complex-plane integral identity from the unified transform method: with $Q$ defined in (2.3), the PDE is written in divergence form, Green’s theorem yields a relation among the Fourier transforms of the initial data, the solution on the interval, and boundary integrals. For each equation, multiplying by $i\\lambda$ and integrating by parts converts the global relation into an equation for the Fourier transform of $q_x$, which is then inverted and evaluated at the two moving endpoints. The result is a system of two linear Volterra equations; the kernels $K_{jm}$ in (1.9) and (1.12)–(1.13) carry the interaction between the endpoints, with the diagonal case $j=m$ weakly singular and the off-diagonal cases regular for heat or regularized by $\\varepsilon$ for Schrödinger. The $\\varepsilon$-regularized kernels and the geometric denominators $H_1,H_2$ are what make the existence proof work.","core_discovery":"The central discovery is that the global relation—an identity obtained by applying Green’s theorem to the divergence form of the PDE—contains enough information to close the Dirichlet-to-Neumann map on a finite time-dependent interval. Evaluating the inverse Fourier transform of the derivative $q_x(x,t)$ at the two endpoints yields a coupled system, equation (1.7) for the heat equation and (1.10) for the linear Schrödinger equation. For heat, the coupling kernels are weakly singular at worst, and classical Volterra theory gives a unique $C^1$ solution. For Schrödinger, the cross kernels are genuinely singular and must be regularized; after an $\\varepsilon$-limit and integration by parts, the paper obtains a generalized Volterra system, equations (4.4)–(4.5), whose kernels are weakly singular provided the geometric quantities $H_1$ and $H_2$ do not vanish, which is guaranteed by the convexity condition (1.14) or by linear boundaries of the form (4.6).","pith_inferences":["An implicit extension is that the same global-relation-plus-Volterra strategy should apply to other linear evolution equations on time-dependent intervals, with the main difficulty shifting to the oscillatory regularization of the cross kernels.","The convexity condition (1.14) is sufficient but likely not necessary; the actual requirement is only that $H_1(t,s)$ and $H_2(t,s)$ stay nonzero, so the theorem could be pushed to broader boundary classes by proving non-vanishing under weaker hypotheses.","A practical consequence the paper does not spell out is numerical: the explicit system (1.7) or (4.4)–(4.5) can be discretized directly, giving a way to compute boundary fluxes on moving domains without resolving the PDE in the interior."],"forward_implications":["For the heat equation on a time-dependent interval, the Dirichlet-to-Neumann map is well defined and unique: the boundary fluxes $q_x(l_1(t),t)$ and $q_x(l_2(t),t)$ are the unique $C^1$ solution of the system (1.7), with no extra geometric condition on the boundaries beyond differentiability and $l_1<l_2$.","For the linear Schrödinger equation, the same conclusion holds for boundary curves satisfying the convexity condition (1.14) or for linear boundaries of the form (4.6), but the paper does not assert it for arbitrary moving boundaries.","Once the Neumann data are obtained, formula (2.9) gives an explicit representation of the solution on the whole domain, and the known uniqueness result for the global relation guarantees that this representation is the actual unique solution of the boundary value problem.","The result extends the earlier half-line analysis to the two-sided moving interval, where the main new feature is a coupled system of two integral equations rather than a single one, and the off-diagonal kernels require regularization in the Schrödinger case."],"supporting_citations":[{"why":"It establishes that if given and reconstructed boundary data satisfy the global relation, the associated boundary value problem has a unique regular solution, turning the Dirichlet-to-Neumann construction into a well-posedness result.","marker":"[Xia(2019)]"},{"why":"It supplies the half-line version of the Volterra existence argument that Theorems 1 and 2 extend to the finite two-sided interval.","marker":"[Fokas-Pelloni(2012)]"},{"why":"It provides the classical existence theory for Volterra integral equations with weakly singular kernels, invoked to prove uniqueness for the heat system and for the regularized Schrödinger system.","marker":"[Miller-Feldstein(1971)]"},{"why":"It supplies the iterative existence argument for the generalized Volterra equations (4.4)–(4.5), whose right-hand sides involve first derivatives of the unknown functions.","marker":"[Brunner(2017)]"}],"fun_headline_variants":["2x2 system yields moving-domain boundary fluxes for heat and Schrödinger","Heat and Schrödinger fluxes on moving domains via 2x2 integral system","Moving intervals: 2x2 system uniquely fixes boundary fluxes","2x2 integral system pins down moving-domain heat and Schrödinger fluxes","Unique 2x2 system recovers boundary fluxes on time-dependent intervals"],"cache_read_input_tokens":17664,"weakest_assumption_plain":"For the Schrödinger theorem, the proof assumes that the solutions of the regularized approximating system converge as the regularization is removed, without actually proving that convergence for the two-equation system.","fun_headline_variants_meta":{"raw":{"variants":["2x2 system yields moving-domain boundary fluxes for heat and Schrödinger","Heat and Schrödinger fluxes on moving domains via 2x2 integral system","Moving intervals: 2x2 system uniquely fixes boundary fluxes","2x2 integral system pins down moving-domain heat and Schrödinger fluxes","Unique 2x2 system recovers boundary fluxes on time-dependent intervals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00114,"raw_usage":{"total_tokens":4706,"prompt_tokens":896,"completion_tokens":3810,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":512,"completion_tokens_details":{"reasoning_tokens":3715}},"tokens_in":512,"tokens_out":3810,"duration_ms":28103,"temperature":1.0,"reasoning_tokens":3715,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:05:08.751820+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a boundary pair satisfying (1.14), for example $l_1(t)=-t+t^2/2$ and $l_2(t)=1+t-t^2/2$ on a small interval, solve the regularized system (4.1) numerically for decreasing $\\varepsilon>0$, and check whether the solutions converge to a limit satisfying (4.4)–(4.5). If the limit fails to exist or violates the equations, the Schrödinger theorem’s proof has a hole; for the heat equation, testing (1.7) against an exact solution with known boundary fluxes would confirm or falsify the uniqueness claim.","supporting_citations":[],"review_version":1}