{"id":"51030792-9d3c-4602-8269-396ead3aed17","arxiv_id":"2501.18228","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"For a time-fractional diffusion model with time-dependent order alpha(t), the space-dependent source is claimed to be uniquely determined by local interior or boundary observations, with Lipschitz stability in a weak norm.","lead":"The paper proves uniqueness and conditional stability for recovering a space-dependent source in a variable-exponent sub-diffusion equation from local interior or boundary measurements. It also demonstrates iterative reconstruction algorithms on smooth and non-smooth sources, though a key mathematical lemma is internally inconsistent.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.1's Duhamel representation is false as stated: substituting u=θ*v into (3.1) with v solving the stated auxiliary problem leaves an extra θ*(g̃f) term, so the proof of Theorem 3 does not establish the central uniqueness claim.","rationale":"The reader's verdict is confirmed. The single most load-bearing concern is the internally inconsistent Duhamel lemma. Substituting u=θ*v into (3.1) yields an uncompensated θ*(g̃f) term because the auxiliary equation for v is nonhomogeneous. This invalidates the representation on which Theorem 3's uniqueness proof and the norm equivalence in Theorem 5 rest. The Titchmarsh step is a secondary issue; even if it were repaired, the false representation still blocks the proof. The paper does contain plausible forward-analyticity arguments and suggestive numerics, but no code or data are supplied, and the central proof is not correct as written. Since the claimed fix is straightforward, a revised version might be viable, but the current manuscript cannot support its main result.","tokens_in":17138,"tokens_out":9945,"duration_ms":89889,"concrete_test":"Independently verify Lemma 3.1 by direct substitution. Define the residual R = c∂^{α0}_t(θ*v) − ∆(θ*v) + g̃′*(θ*v) − βf. Using v's stated equation, compute R = θ*(g̃f). For a concrete numerical check, take α(t)=0.5+0.25t, f≡1, β≡1 on [0,1]^2, solve (3.1) and the stated auxiliary problem with a finite-element method, and compare u against θ*v; the difference will be nonzero and proportional to θ*(g̃f), confirming the inconsistency.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3.1 claims u=θ*v solves (3.1) when J^{1−α0}θ=β and v solves c∂^{α0}_t v−∆v+g̃′*v = g̃f with v(0)=f. Direct substitution: c∂^{α0}_t u−∆u+g̃′*u = J^{1−α0}θ(t)v(0) + θ*(c∂^{α0}_t v−∆v+g̃′*v) = βf + θ*(g̃f). The stated auxiliary right-hand side g̃f is not zero, so the extra convolution term does not vanish; the representation does not solve (3.1). This is the exact step that makes the proof of Theorem 3 work: the observation u=0 on Ω0×(0,T) is converted to β*v=0, and Titchmarsh's theorem is applied to infer v=0 and hence f=0. With Lemma 3.1 false, that chain is broken. The correction is simple—set the auxiliary right-hand side to zero—but the manuscript as submitted contains a false lemma at the core of its main theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the variable-exponent sub-diffusion model (1.1). It first converts the equation to an equivalent constant-exponent form with a convolution perturbation (Lemma 2.1), proves time-analyticity of the solution (Lemma 2.4), and derives a weak unique continuation principle (Theorems 1–2). These tools are then applied to the inverse space-dependent source problem: Theorem 3 claims uniqueness of the source f from interior observations u|_{\\Omega_0\\times(0,T)} under a nonzero temporal factor \\beta(t), Theorem 4 claims the analogous result from boundary flux data, and Theorems 5–6 establish conditional Lipschitz stability in specially defined weak norms. A numerical section tests iterative reconstruction algorithms for smooth and non-smooth sources.","tokens_in":17338,"tokens_out":12557,"duration_ms":110578,"significance":"If the results hold, the paper would make a useful contribution: it extends inverse-source uniqueness and stability theory to sub-diffusion models with time-dependent fractional order, using a tractable perturbation transformation and analytic-continuation techniques. The weak-norm stability estimates are of interest, and the numerical experiments, while heuristic, illustrate the feasibility of the proposed reconstruction approach. However, the central proof currently rests on a false lemma and a formally invalid application of Titchmarsh's theorem, so the theoretical claims are not established in the submitted version.","major_comments":[{"comment":"The auxiliary problem in Lemma 3.1 is stated with right-hand side g̃f, but substituting u = θ*v into (3.1) with v solving that problem yields c∂^{α0}_t u − ∆u + g̃′*u = β(t)f(x) + (θ*(g̃f))(t), because J^{1−α0}θ = β, v(0)=f, and the differential expression applied to v equals g̃f, not zero. Thus the claimed Duhamel representation does not solve (3.1) unless g̃f ≡ 0. The correct statement should take v to solve the homogeneous equation c∂^{α0}_t v − ∆v + g̃′*v = 0 with the same initial and boundary conditions. Since Theorem 3 and the subsequent stability results rely directly on this lemma, the error is load-bearing and must be fixed.","section":"Section 3.1, Lemma 3.1"},{"comment":"After deriving 0 = β*v on Ω0×(0,T), the paper invokes Titchmarsh's convolution theorem to conclude that v vanishes on all of Ω0×(0,T). The finite-interval form of Titchmarsh's theorem only ensures that there exist a,b ≥ 0 with a+b ≥ T such that β = 0 a.e. on (0,a) and v = 0 a.e. on (0,b); it does not imply v = 0 on the full interval. To make the argument valid, the authors must use the t-analyticity of v supplied by Lemma 2.4 to extend from the subinterval to (0,T). As written, this step is not justified.","section":"Section 3.1, proof of Theorem 3"},{"comment":"The proof introduces η = s^{α0} + s g̃(s) and asserts that the identity Σ (u0,φn)φn/(λn+η) = 0 holds for η in an open set O, but it does not show that the image of the relevant s-domain under η contains an open set. Since η is analytic and nonconstant, this follows from the open mapping theorem after a short argument (e.g., g̃(s)→0 as |s|→∞ while s^{α0} is nonconstant), but the paper omits this justification. Additionally, the final reduction to u0 = 0 is only cited to [19]; it should be stated in enough detail for the reader to verify the argument.","section":"Section 2.3, proof of Theorem 1"}],"minor_comments":[{"comment":"The section heading contains a typo: 'well-posdeness' should be 'well-posedness'.","section":"Section 2.1 (title)"},{"comment":"The sentence 'Notice the zero initial condition' is misleading; the boundary term in the integration by parts vanishes because g̃(0)=0, not because the initial value u0 is zero.","section":"Lemma 2.1 proof"},{"comment":"In the displayed equation following the Laplace transform, a closing bracket is missing: the right-hand side should read (s^{α0−1} + g̃(s))u0.","section":"Section 2.3, Laplace-transformed equation"},{"comment":"The symbol β is used for a power bound in the statement of the corollary, while Section 3 uses β(t) for the temporal factor of the source; this notation clash should be resolved (e.g., by using ρ or γ for the power).","section":"Corollary 2.3"},{"comment":"The final sentence of the proof reads 'We finish the proof of the lemma' but should be 'the theorem'.","section":"Theorem 3 proof"},{"comment":"There are small language errors: 'date-match functional' should be 'data-match functional', and 'characterization function' should be 'characteristic function'.","section":"Section 4"},{"comment":"The existence of θ satisfying J^{1−α0}θ = β for arbitrary β ∈ W^{1,1}(0,T) is asserted without proof or sufficient regularity conditions; a brief justification (e.g., taking θ as an appropriate fractional derivative of β) should be added.","section":"Lemma 3.1"}],"recommendation":"major_revision","confidential_remarks":"The central claims are likely salvageable: the Duhamel principle is standard once the auxiliary right-hand side is corrected to zero, and the Titchmarsh gap can be closed using the already-established analyticity of v. However, the submitted manuscript contains a false lemma at the core of the main theorem and a formally invalid step in the proof of Theorem 3. The paper also relies heavily on self-citations [17] and [29] for well-posedness and operator estimates, which is acceptable if those results are precisely invoked, but the inverse-source results are new. I recommend major revision rather than rejection because the load-bearing errors appear correctable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The genuinely new content is the analytic extensibility and weak unique continuation for the variable-exponent sub-diffusion model, and the inverse-source uniqueness and stability that follow from them. That is a real step for this niche. But the proof rests on a Duhamel lemma that is false as stated, and the fix is not cosmetic: Theorem 3 and the norm property behind Theorem 5 both lean on it.\n\nWhat the paper does well: the perturbation reduction to constant order is borrowed from the authors' own earlier work, but the use of t-analyticity plus Laplace transform to get unique continuation is a reasonable route, and the inverse-source statements themselves are not in the cited predecessors. The stability norm is weak but honestly described. The numerics are standard iterative regularization, with no benchmark claims; they are adequate as an illustration.\n\nWhere the soft spots are. Lemma 3.1 claims u=theta*v solves (3.1) when v solves the auxiliary problem with right-hand side g-tilde f. Direct substitution leaves an extra theta*(g-tilde f) term, so the representation does not satisfy the equation. The repair is simple - set the auxiliary right-hand side to zero, so v solves the homogeneous equation with initial data f - but as submitted the central uniqueness proof does not work. A second issue: the Titchmarsh step concludes v=0 on the whole interval from beta*v=0, but the finite-interval theorem only gives vanishing on a subinterval unless beta has no initial gap; an analyticity argument would close that, and the ingredients are already in the paper. There is also a typo in the Laplace-domain display (an extra s^{alpha0}), and the numerical section ships no code or data, though the examples are enough to show feasibility. The self-citations are not circular; they supply the perturbation framework, and the inverse results do not reduce to them.\n\nMy take: the idea is plausible and probably repairable, but the manuscript's main theorem is not supported as written. This deserves referee time - a good referee can point to Lemma 3.1 and the Titchmarsh gap quickly. I would send it to review and expect major revision.","headline":"The inverse-source uniqueness idea is new and probably repairable, but the submitted proof of the central theorem rests on a false Duhamel lemma and should go back for major revision.","tokens_in":17924,"tokens_out":5235,"would_cite":false,"duration_ms":46038,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R11","35R30","65M32"],"pacs":[],"model":"deepseek-v4-flash","headline":"Local time-series data uniquely fix a diffusion source","keywords":["inverse source problem","variable-exponent sub-diffusion","weak unique continuation","perturbation method","fractional diffusion","Lipschitz stability","iterative regularization"],"falsifier":"Substitute $u=\\theta*v$ into the first equation of (3.1) and use the auxiliary equation defining $v$; the left side should reduce to $\\beta f$, but a direct computation gives $\\beta f+\\theta*(\\tilde g f)$. Whether this extra term is nonzero for a non-constant $\\alpha(t)$ and a nonzero $f$, checked symbolically or numerically, settles Lemma 3.1 as stated.","tokens_in":16837,"feed_emoji":"⏳","tokens_out":10874,"duration_ms":87667,"temperature":0.7,"pith_summary":"This paper studies the variable-exponent sub-diffusion equation, in which the fractional order $\\alpha(t)$ changes with time, and asks whether the space-dependent source $f(x)$ can be recovered from observing the solution on a small subdomain over time. The main claim is uniqueness: if the observed solution vanishes on $\\Omega_0\\times(0,T)$, then $f$ must vanish everywhere in $\\Omega$, provided the known time factor $\\beta$ is nonzero and integrably differentiable. The paper also proves a conditional Lipschitz stability estimate in a specially designed weak norm, meaning small data errors produce small source errors in that norm. The route is a perturbation method that rewrites the variable-exponent model as a constant-order fractional equation with a convolution correction, followed by analytic continuation, a weak unique-continuation argument, and a variational identity with the adjoint problem. Numerical experiments with iterative regularization show smooth and piecewise-constant sources reconstructed from local interior data.","feed_headline":"Local time-series data uniquely fix a diffusion source","feed_subtitle":"A variable-exponent sub-diffusion model needs only a small interior probe to recover the full spatial source.","key_machinery":"The central device is the perturbation method, which rewrites the variable-exponent kernel $t^{-\\alpha(t)}/\\Gamma(1-\\alpha(t))$ as the constant-order kernel $t^{-\\alpha_0}/\\Gamma(1-\\alpha_0)$ plus a correction $\\tilde g(t)$, turning (1.1) into the constant-order equation $c\\partial^{\\alpha_0}_t u-\\Delta u+\\tilde g'*u=\\tilde g u_0+F$. This makes the solution analytically extendable beyond $T$, so Laplace transforms and eigenfunction expansions apply. The uniqueness argument then uses Duhamel's representation $u=\\theta*v$ with $J^{1-\\alpha_0}\\theta=\\beta$ and $v$ solving an auxiliary problem with initial value $f$; the weak unique-continuation principle forces $v\\equiv 0$, hence $f=0$. Stability is carried by the variational identity $\\langle \\beta f,\\varphi[\\omega]\\rangle=\\langle u[f],\\omega\\rangle$ connecting the inversion data to a weak norm $\\|\\cdot\\|_B$ built from adjoint solutions.","core_discovery":"On its own terms, the paper establishes that for the zero-initial-data problem $c\\partial^{\\alpha_0}_t u-\\Delta u+\\tilde g'*u=f(x)\\beta(t)$ with zero boundary data, measuring $u$ on any nonempty subdomain $\\Omega_0$ over the whole time interval determines $f$ uniquely: Theorem 3 states that if $\\beta\\ne 0$ and $\\beta\\in W^{1,1}(0,T)$, then $u=0$ on $\\Omega_0\\times(0,T)$ implies $f=0$ in $\\Omega$. Theorem 4 gives the same conclusion from vanishing Neumann data on a subboundary. The stability result (Theorem 5) bounds the difference of two sources in the weak norm $\\|\\cdot\\|_B$ by the $L^1$ difference of their interior data, and Theorem 6 does the same for flux data. The proof chain relies on extending solutions analytically in time to an infinite interval, applying the Laplace transform to get a weak unique-continuation principle, and using Duhamel's principle to carry the observation from $u$ to an auxiliary solution $v$ whose initial value is $f$.","pith_inferences":["Because the uniqueness proof depends on Lemma 3.1's Duhamel representation, the first check for a reader should be whether substituting $u=\\theta*v$ into (3.1) leaves an extra term $\\theta*(\\tilde g f)$; if it does, the theorem needs a revised auxiliary problem rather than the one stated.","The analytic-extensibility result suggests a stronger inverse statement than the paper proves: observations on a short time interval, or at finitely many times, may already determine $f$, since analyticity in $t$ propagates local time data.","The weak norm $\\|\\cdot\\|_B$ is defined through adjoint solutions and is not what the numerical algorithms minimize; connecting the stability guarantee to the $L^2$-Tikhonov and TV functionals actually used is an open step."],"forward_implications":["A single interior observation over time is, in principle, enough to determine the whole spatial source; no boundary or full-domain measurement is required.","The same uniqueness holds from flux measurements on an open part of the boundary.","Noise in the data propagates linearly: the source error in the weak $B$-norm is at most a constant times the $L^1$ data error.","Analytic extensibility of solutions brings Laplace-transform and eigenfunction tools to variable-exponent models, matching the toolkit used for constant-order sub-diffusion.","The iterative thresholding and Nesterov-type schemes reconstruct smooth and non-smooth sources from local data in the numerical tests."],"supporting_citations":[{"why":"supplies the perturbation method that rewrites the variable-exponent model as a constant-order equation with a convolution correction; the paper's analysis builds on this equivalence.","marker":"[29]"},{"why":"provides the estimates (7) and the iterative analytic-extension argument used to extend solutions to an infinite time interval.","marker":"[17]"},{"why":"supplies the eigenfunction-series argument used to conclude $u_0=0$ from vanishing data on a subdomain.","marker":"[19]"},{"why":"gives the weak unique-continuation framework for fractional diffusion-advection equations that the paper adapts to the variable-exponent setting.","marker":"[11]"},{"why":"supplies the fractional calculus identities, including the integration-by-parts formula in Lemma 3.2, behind the variational identity.","marker":"[8]"},{"why":"introduces the weak-norm stability approach adapted here for the interior-observation inverse source problem.","marker":"[24]"},{"why":"provides the distributed-order inverse-source template for the adjoint-based weak norm and bilinear form.","marker":"[25]"}],"fun_headline_variants":["Local probe uniquely recovers sub-diffusion source","Sub-diffusion source fixed by interior time data alone","One subregion's data pins down the diffusion source","Variable-exponent model: local data yield unique source","Small interior observation suffices for source uniqueness"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Lemma 3.1's Duhamel representation $u=\\theta*v$, with $v$ solving the auxiliary problem whose forcing is $\\tilde g f$; if the representation actually produces an extra convolution term $\\theta*(\\tilde g f)$, the uniqueness and stability results do not follow from the proof as written.","fun_headline_variants_meta":{"raw":{"variants":["Local probe uniquely recovers sub-diffusion source","Sub-diffusion source fixed by interior time data alone","One subregion's data pins down the diffusion source","Variable-exponent model: local data yield unique source","Small interior observation suffices for source uniqueness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000163,"raw_usage":{"total_tokens":1228,"prompt_tokens":912,"completion_tokens":316,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":528,"completion_tokens_details":{"reasoning_tokens":241}},"tokens_in":528,"tokens_out":316,"duration_ms":3754,"temperature":1.0,"reasoning_tokens":241,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T00:15:52.933021+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Substitute $u=\\theta*v$ into the first equation of (3.1) and use the auxiliary equation defining $v$; the left side should reduce to $\\beta f$, but a direct computation gives $\\beta f+\\theta*(\\tilde g f)$. Whether this extra term is nonzero for a non-constant $\\alpha(t)$ and a nonzero $f$, checked symbolically or numerically, settles Lemma 3.1 as stated.","supporting_citations":[{"cited_title":"Zheng, Two methods addressing variable-exponent fr actional initial and boundary value problems and Abel integral equation","cited_arxiv_id":null,"evidence_quote":"supplies the perturbation method that rewrites the variable-exponent model as a constant-order equation with a convolution correction; the paper's analysis builds on this equivalence."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the estimates (7) and the iterative analytic-extension argument used to extend solutions to an infinite time interval."},{"cited_title":"Sakamoto and M","cited_arxiv_id":null,"evidence_quote":"supplies the eigenfunction-series argument used to conclude $u_0=0$ from vanishing data on a subdomain."},{"cited_title":"Jiang, Z","cited_arxiv_id":null,"evidence_quote":"gives the weak unique-continuation framework for fractional diffusion-advection equations that the paper adapts to the variable-exponent setting."},{"cited_title":"Sun and J","cited_arxiv_id":null,"evidence_quote":"introduces the weak-norm stability approach adapted here for the interior-observation inverse source problem."},{"cited_title":"Sun and J","cited_arxiv_id":null,"evidence_quote":"provides the distributed-order inverse-source template for the adjoint-based weak norm and bilinear form."}],"review_version":1}