{"id":"399c2aeb-bbdc-4cdd-ae79-75cc3459e460","arxiv_id":"2501.11332","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Claims existence, uniqueness, and stability for inverse Stefan source and flux recovery, but the proof omits coupling terms and is circular in the source definition.","lead":"The paper claims to prove existence, uniqueness, and stability for four inverse Stefan problems that recover a time-dependent heat source or boundary heat flux. The key series construction drops mode-coupling terms and treats the unknown source as known data, so the main theorems are not supported.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The modal series (23) does not solve the PDE (11): the off-diagonal advection terms ∫ b φ'_m φ_n dξ with b=ξs'/s are nonzero for m≠n, so the diagonal system (21) is not equivalent to the weak formulation (17).","rationale":"The reader's weakest assumption identifies exactly the load-bearing flaw: the Fourier decoupling that produces the independent ODEs (21) ignores the off-diagonal advection coupling in (17). I independently checked the relevant matrix elements for b(ξ,t)=ξs'/s and found them nonzero; for m=1,n=2 the value 4/3 is explicit. This makes the construction (23) fail to satisfy the stated PDE, and no later estimate repairs the gap, since the lemmas only prove norm convergence, not equation satisfaction. The circularity of h depending on the unknown R is a second serious defect, and the denominator bounds in Theorem 4 are also not justified, but the modal decoupling is the more fundamental issue because it invalidates the representation underlying every existence and uniqueness theorem in the paper. I therefore agree with the reader's REJECT verdict; no verdict adjustment is needed.","tokens_in":30608,"tokens_out":4762,"duration_ms":50732,"concrete_test":"Analytically test the claimed decoupling at t=0 with s(t)=1+t, so s(0)=1, s'(0)=1, b(ξ,0)=ξ, a(0)=1. Choose R(t)=0 and initial data ~ψ(ξ)=φ_2(ξ)=√2 sin(2πξ). The series (23) then gives U(t)=e^{-∫_0^t(aλ_2-b_diag)dτ}φ_2, with b_diag=∫ξ φ'_2 φ_2 dξ=-1/2. Evaluate the m=1 weak-form residual of (11) at t=0: ⟨U_t-U_ξξ-ξU_ξ,φ_1⟩ = ∫_0^1 (1/2 φ_2-ξφ'_2)φ_1 dξ = -∫_0^1 ξ φ'_1 φ_2 dξ = -4/3 ≠ 0. If this entry is nonzero, the residual is nonzero and the function (23) is not a weak solution of (11); this directly disproves the equation-satisfaction assertion of Lemma 2 and the existence claim of Theorem 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result, Theorem 1, asserts existence and uniqueness of a weak solution pair (R,U) with U given by (23) and R by (26). The derivation of (23) decouples the Fourier modes: after expanding U=Σ U_n φ_n, the paper keeps only the m=n equations (20)-(21) and discards the m≠n case in (18)-(19). But this decoupling is not valid for the advection term −b(ξ,t)Uξ with b(ξ,t)=ξs'(t)/s(t). In the m-th weak equation the advection contribution is Σ_n U_n ∫ b φ'_n φ_m dξ, and these off-diagonal integrals are not zero. For example, with s(t)=1+t at t=0 (so b=ξ), m=1, n=2, one computes ∫_0^1 ξ φ'_1 φ_2 dξ = 4/3 ≠ 0. Consequently a U built from independently evolved Fourier coefficients, as in (23), does not satisfy the full weak formulation (17): the off-diagonal terms produce a nonzero residual. Lemma 2 does not repair this; its proof only establishes that the series (23) belongs to certain function spaces via Bessel and Cauchy-Schwarz estimates (27)-(33), and never verifies that (23) satisfies the PDE. Thus existence in Theorem 1 is not established, and the uniqueness argument built on the same representation also collapses. A further independent obstruction is circularity: in (9), h(x,t)=f+u*s'/(Rs^2)x contains the unknown R(t) in the denominator, so the Fourier coefficients ~h_n(t) used in (23) and (26) are not known data; (26) is therefore not an explicit Volterra equation for R. These are load-bearing gaps in Theorems 1, 2, 5, 8, and 11.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies four inverse Stefan problems for the one-dimensional heat equation on the moving domain (0,s(t)). In each case the unknowns include a time-dependent source coefficient (R(t) or P(t)) and, for the Neumann problems, the boundary heat flux q(t). The authors transform each problem to a fixed domain, expand the solution in the eigenfunctions of the Dirichlet or Neumann spectral problem, and derive explicit Fourier series representations for U together with Volterra-type equations for the unknown coefficient R or q. The main claims are existence, uniqueness, and continuous dependence of the weak solution pair under regularity and sign assumptions (A1)-(A3) or (B1)-(B4).","tokens_in":31000,"tokens_out":7839,"duration_ms":74249,"significance":"If the central construction were correct, the paper would supply explicit spectral formulas for recovering time-dependent source coefficients and heat fluxes in Stefan problems, together with a systematic continuous-dependence framework; the presentation of four problem formulations in a unified way is also useful. However, the derivation rests on a spectral decoupling that is not valid for the advection term in the transformed equation, and the recovery formulas for R depend on transformed source functions that themselves contain R. As a result, the main theorems do not currently establish the announced results.","major_comments":[{"comment":"The reduction from the weak formulation (17) to the diagonal system (21) is invalid. The advection contribution in the m-th equation is Σ_n U_n(t) ∫_0^1 b(ξ,t) φ'_n(ξ) φ_m(ξ) dξ with b(ξ,t)=ξs'(t)/s(t), and these integrals do not vanish for m≠n; for instance, with s(t)=1+t at t=0 and b(ξ)=ξ, ∫_0^1 ξ φ'_1(ξ) φ_2(ξ) dξ = 4/3 ≠ 0. Hence the modal series (23) built from the independent equations (22) does not satisfy the full weak formulation (17), and the paper provides no argument that the discarded m≠n equations are satisfied. Lemma 2 does not repair the gap: its proof only establishes norm bounds for the series (23) and never verifies the PDE (11) or the weak form (17).","section":"§2, Eqs. (17)–(21)"},{"comment":"The recovery formula (26) is not a closed Volterra equation for R(t). The transformed source is defined in (9) by h(x,t)=f(x,t)+u_* s'(t)/(R(t)s²(t))x, so the Fourier coefficients ~h_n(t) used in (23) and (26) depend on the unknown R(t) itself. Thus R appears nonlinearly on both sides of (26), including in the denominator w(t)=Σ(-1)^n√λ_n ~h_n(t), and the assertion in Theorem 1(iii) that R is 'defined by (26)' is circular. The same circularity affects the corresponding formulas in Sections 3–5 and the uniqueness arguments built on them.","section":"§2, Eqs. (9) and (26)"},{"comment":"The proof of Lemma 2 does not demonstrate that the function (23) is a weak solution of the problem: it only shows that the series belongs to C([0,T];L²(Ω₀)) and C([0,T];H²₀(Ω₀)) under certain summability assumptions, without checking the equation, the boundary conditions, or the initial condition. Moreover, estimate (35) for U_t involves R'(t), while R is only assumed to be in C[0,T] in Theorem 1(iii); no higher regularity of R is established before it is used in the proof.","section":"§2, Lemma 2 and Theorem 1"},{"comment":"The assumptions on the given data are stated in terms of the unknown solution. For example, (A3)1 requires f(s(t),t) = -u_* s'(t)/(R(t)s(t)) and f'(0,t)=f'(s(t),t)=-u_* s'(t)/(R(t)s²(t)), so the admissibility of the data depends on the coefficient R(t) that is to be determined. This makes the hypotheses not checkable from the data alone and compounds the circularity of the recovery formula.","section":"§2, hypotheses (A3) and (B3)"},{"comment":"The uniqueness proof in Theorem 1 contains a vacuous identity: equation (38) states Û - Ũ = Σ ∫_0^t (R̂(τ) - R̂(τ)) ̃f_n(τ) ... dτ φ_n(ξ), whose integrand vanishes identically. The subsequent conclusion that R̂ = R̃ from (39) therefore does not follow from the written formulas.","section":"§2, Eq. (38)"}],"minor_comments":[{"comment":"The initial condition after the transformation is written as ψ(x)=φ(x)-u_*/s(t)x; it should be evaluated at t=0, i.e., with s(0).","section":"§2, Eq. (9)"},{"comment":"The term +a(t)λ_n(∂ξξU,φ_m) is not the correct weak form of -a(t)∂ξξU; after integration by parts the eigenvalue enters as a factor of (U,φ_m), not of (∂ξξU,φ_m).","section":"§2, Eq. (17)"},{"comment":"In the proof of Lemma 1, the function g is defined as -~ψ'''_n(ξ), but the subscript n should be removed; it is the third derivative of ~ψ, not of the Fourier coefficient.","section":"§2, Lemma 1 proof"},{"comment":"The claimed bound (36) is dimensionally inconsistent unless additional summability of 1/λ_n is used; the constant C8 is not specified.","section":"§2, Lemma 3"},{"comment":"The statement of Theorem 2 contains the apparent typo 'L2(0,T;L2(0,Ω))', which should presumably be L²(0,T;L²(Ω)).","section":"§2, Theorem 2"},{"comment":"The reference [Evan(2010)] should be Evans, Partial Differential Equations, and the year of [Dragomir(2022)] in the reference list is inconsistent with the in-text citation.","section":"References"}],"recommendation":"reject","confidential_remarks":"The errors identified in the main report are load-bearing and not local: the entire spectral method depends on decoupling the Fourier modes, and the circular definition of ~h affects every recovery formula. I therefore recommend rejection. I would also note that the manuscript shows numerous signs of incomplete editing, including inconsistent assumptions, repeated symbols with different meanings, and typos in displayed equations, which would require a full rewrite even if the mathematical gaps were resolved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Honest take: this paper has the right shape for a useful contribution but the central machinery doesn't close. The authors take a known spectral method for inverse parabolic problems, adapt it to four Stefan-type inverse problems, and state existence/uniqueness/stability theorems for each. What's genuinely new is the specific combination: moving boundary, time-dependent source coefficient R(t), and recovery of boundary heat flux q(t) in the Neumann cases. If the estimates were valid, the Volterra formulation of R(t) would be a sensible route.\n\nThe basic setup is fine: fix the domain by ξ = x/s(t), expand in the eigenbasis of −φ'' = λφ with the relevant boundary conditions, derive ODEs for the Fourier coefficients, and close the system with the Stefan condition. The authors also include maximum-principle-inspired stability estimates and honestly cite the prior work they build on (Ismailov–Ozawa–Suragan, Kerimov–Ismailov, Ivanchov). That part is legitimate and shows they know the literature.\n\nThe problems are in the load-bearing steps. In Section 2, the variational system (17) with b(ξ,t)=ξs'/s couples all modes: the advection term ∫ b φ'_m φ_n dξ is not zero for m≠n (e.g., s(t)=1+t, t=0, m=1, n=2 gives 4/3). Dropping those off-diagonal equations after saying R(t) is not unique is not a proof tactic; the series (23) built from diagonal ODEs does not satisfy the PDE (11). Lemma 2 only proves norm convergence of the series, not that it satisfies the equation or the weak form. Second, the source term h(x,t)=f+u* s'/(R s^2) x contains R(t) in the denominator, so ~h_n(t) is not known data when you're solving for R. The recovery formula (26) treats ~h as input, which is circular. Equation (26) is not an explicit Volterra equation for R; R sits on both sides through the definition of ~h. The same problems recur in Theorems 5, 8, and 11, which just repeat the same derivation for different boundary conditions. The continuity estimates (Theorems 3, 6, 9, 10) are appended rather than derived from the constructed solution, and they assume a maximum principle that doesn't apply to weak solutions of the transformed equation with the sign conditions stated.\n\nSo who is this for? Someone working on inverse Stefan problems might find the formal setup worth reading, but only as a starting point for repair. As submitted, the central results are not established. The paper deserves a serious referee because the questions are real and the flaws are subtle enough that expert judgment is needed. My recommendation: send it to peer review, but expect a reject-and-resubmit with major revisions. I would not cite it in its current form.","headline":"A spectral-formal treatment of four inverse Stefan problems whose central mode decoupling is invalid and whose recovery formula is circular; the combination is new but the theorems are not established.","tokens_in":31546,"tokens_out":3668,"would_cite":false,"duration_ms":33464,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","35R35","35K20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that four inverse Stefan problems—recovering a time-dependent source coefficient and, in Neumann cases, the heat flux—each reduce to a Volterra integral equation with existence, uniqueness, and continuous dependence.","keywords":["inverse Stefan problem","time-dependent source coefficient","heat flux recovery","weak solution","spectral method","Volterra integral equation","continuous dependence","phase-change heat conduction"],"falsifier":"Compute the projection integrals ∫$_0^{1}$ ξ φ_m'(ξ) φ_n(ξ) dξ for the sine basis; for m≠n they are nonzero, so (b Uξ,φ_m) contains cross terms B_{mn}(t)U_n(t). Evaluating the residual of (11) with U from (23) at a time when s'(t)≠0 decides whether the paper's constructed solution is actually a weak solution.","tokens_in":30308,"feed_emoji":"🔥","tokens_out":5732,"duration_ms":56555,"temperature":0.7,"pith_summary":"The paper claims that a moving-boundary Stefan problem can be solved backwards: from boundary data and the known phase-change interface, one can recover the time-dependent source coefficient R(t) (or P(t)) and, in the Neumann cases, the heat flux q(t). The authors transform the moving domain to a fixed interval, expand the temperature in eigenfunctions of that interval, and reduce each inverse problem to a scalar Volterra integral equation. If the theorems are right, existence, uniqueness, and continuous dependence on the data hold for all four problem formulations. The practical stake is that heat-source strength in phase-change materials could be identified from ordinary boundary measurements.","feed_headline":"Inverse Stefan source terms recovered by spectral series","feed_subtitle":"Paper reduces four moving-boundary heat problems to Volterra equations, with uniqueness and continuous dependence.","key_machinery":"The machinery is a Fourier expansion in the Dirichlet eigenfunctions φ_n=√2 sin(nπξ) or the Neumann eigenfunctions φ_n=√2 cos((2n-1)πξ/2) of the fixed interval (0,1), reached by the change of variables ξ=x/s(t). The load-bearing reduction is the claim in (20)-(21) that the moving-boundary term (b(ξ,t)Uξ,φ_n) equals b(t)U_n(t) with the same index n, so the PDE becomes an infinite set of decoupled ordinary differential equations for the Fourier coefficients U_n(t). Solving those ODEs turns the Stefan boundary condition into a series; differentiating it in t produces the Volterra integral equation (26) for R(t), and the same route yields q(t) from (62). The explicit identities for the eigenfunctions and the coefficient estimates in Lemmas 1-4 are what make the series and the integral equation convergent.","core_discovery":"The central claim is that the weak solution pair (R,U) of the transformed inverse Stefan problem exists and is unique: U is given by the Fourier series (23), and R(t)>0 is determined by the Volterra equation (26). In the Neumann formulation the unknown heat flux q(t) is recovered from an explicit series formula (62) once R(t) is known, and in the two cases with source term P(t)u+f the substitution R(t)=exp(-∫P) converts a linear-in-u source into the same Volterra structure. The paper further claims continuous dependence of U and R (or q) on the data functions ψ, h, s, c, with explicit estimates in Theorems 3, 4, 6, 7, 9, and 10. If correct, this gives a parameter-free recovery procedure for time-dependent source coefficients in phase-change heat conduction.","pith_inferences":["The off-diagonal mode-coupling check is the natural next step: if it fails, the Volterra equation (26) would describe a diagonalized problem rather than the original PDE, and the recovery claims would need extra terms.","Because the paper's formulas are explicit, numerical implementation is immediate: truncate the series, solve the Volterra equation by quadrature, and compare the resulting R(t) and U(ξ,t) against direct Stefan simulations for slowly moving interfaces.","The same spectral ansatz could be tried for other boundary conditions or higher dimensions, but any such extension inherits the diagonal-mode assumption, which would need separate verification."],"forward_implications":["In the Dirichlet case, the interface condition alone determines R(t) through the Volterra equation (26), without needing extra boundary measurements.","In the Neumann case, the heat flux q(t) on the left boundary is recovered from an explicit series once R(t) is known, so one can infer both the source strength and the boundary flux from the same Stefan condition.","The linear source case P(t)u+f reduces to the R(t)f case by R(t)=exp(-∫P), so existence and uniqueness for P(t) follow without a separate analysis.","The continuous-dependence estimates imply that small errors in the measured data (initial temperature, source shape, interface speed) cause small changes in the recovered coefficient in the sup norm."],"supporting_citations":[{"why":"Supplies the model problem of identifying a time-dependent source coefficient for heat-type equations, which the paper extends to the Stefan setting.","marker":"[IsmOzaSur(2024)]"},{"why":"Provides the general theory of inverse problems for parabolic equations that frames the existence, uniqueness, and dependence analysis.","marker":"[Ivanchov(2003)]"},{"why":"Gives an inverse time-dependent source problem for the heat equation with nonlocal boundary conditions, a benchmark for the spectral method used here.","marker":"[HazLesIsm(2019)]"},{"why":"Establishes the classical result of recovering a time-dependent coefficient in a parabolic equation, which the paper adapts to a moving boundary.","marker":"[CannonRun(1991)]"},{"why":"Treats a two-phase inverse Stefan problem, providing the directly relevant Stefan-problem context that the paper's four formulations build on.","marker":"[KasSur(2023)]"},{"why":"Supplies the maximum-principle arguments used in the continuous-dependence estimates.","marker":"[Evan(2010)]"},{"why":"Provides the Gronwall-type inequality used to turn difference estimates into continuous-dependence bounds for R(t) and q(t).","marker":"[Dragomir(2022)]"}],"fun_headline_variants":["Spectral series solve inverse Stefan source recovery","Inverse Stefan: source and flux from boundary data","Volterra equations unlock inverse Stefan parameters","Unique weak solution for inverse Stefan problem","Time-dependent Stefan sources recovered uniquely"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that the convection term ξ s'(t)/s(t) Uξ does not mix Fourier modes, so that the diagonal equations (21) describe the full solution; if the off-diagonal mode couplings are nonzero, the series (23) built from independent modes need not solve the PDE (11).","fun_headline_variants_meta":{"raw":{"variants":["Spectral series solve inverse Stefan source recovery","Inverse Stefan: source and flux from boundary data","Volterra equations unlock inverse Stefan parameters","Unique weak solution for inverse Stefan problem","Time-dependent Stefan sources recovered uniquely"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000153,"raw_usage":{"total_tokens":1149,"prompt_tokens":831,"completion_tokens":318,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":447,"completion_tokens_details":{"reasoning_tokens":254}},"tokens_in":447,"tokens_out":318,"duration_ms":3370,"temperature":1.0,"reasoning_tokens":254,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T18:23:35.344388+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the projection integrals ∫$_0^{1}$ ξ φ_m'(ξ) φ_n(ξ) dξ for the sine basis; for m≠n they are nonzero, so (b Uξ,φ_m) contains cross terms B_{mn}(t)U_n(t). Evaluating the residual of (11) with U from (23) at a time when s'(t)≠0 decides whether the paper's constructed solution is actually a weak solution.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the model problem of identifying a time-dependent source coefficient for heat-type equations, which the paper extends to the Stefan setting."},{"cited_title":"Ivanchov","cited_arxiv_id":null,"evidence_quote":"Provides the general theory of inverse problems for parabolic equations that frames the existence, uniqueness, and dependence analysis."},{"cited_title":"Hazanee, D","cited_arxiv_id":null,"evidence_quote":"Gives an inverse time-dependent source problem for the heat equation with nonlocal boundary conditions, a benchmark for the spectral method used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the classical result of recovering a time-dependent coefficient in a parabolic equation, which the paper adapts to a moving boundary."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Treats a two-phase inverse Stefan problem, providing the directly relevant Stefan-problem context that the paper's four formulations build on."},{"cited_title":"Evans, Partial Differential equation, AMS, Providence, 2010","cited_arxiv_id":null,"evidence_quote":"Supplies the maximum-principle arguments used in the continuous-dependence estimates."},{"cited_title":"Dragomir, Some Gronwall Type Inequalities and Applications, RGMIA Monographs, Victoria University, Australia, 2002","cited_arxiv_id":null,"evidence_quote":"Provides the Gronwall-type inequality used to turn difference estimates into continuous-dependence bounds for R(t) and q(t)."}],"review_version":1}