{"id":"a202bcb5-30df-4356-998a-337406b84e10","arxiv_id":"2411.14382","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For heat equations with memory, sampling observability at finitely many instants holds exactly when backward uniqueness holds at those instants and the memory kernel is nonzero at one of them.","lead":"This paper asks when one can recover the initial state of a heat equation with memory from measurements made at a few instants and on small regions. The answer is a sharp condition relating the chosen instants, the memory kernel, and a backward uniqueness property, together with a recipe for choosing the instants.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main theorems depend on Lemma 2.1, an asymptotic expansion quoted from [17] under analytic kernels and extended in Remark 2.2 to all C^2 kernels without a supplied proof; if this extension fails, Theorems 1.7 and 1.9 lose their foundation.","rationale":"The reader's weakest-assumption analysis identified exactly the same load-bearing concern: the asymptotic expansion Lemma 2.1 is imported from the authors' prior work under an analyticity assumption and extended to C^2 in Remark 2.2 without a proof. My independent reading of the proof structure confirms that this expansion is not an auxiliary technicality but the mechanism through which the observation regions couple to the memory kernel: the geometric condition (1.4) only becomes visible in the observability inequality (2.5) after replacing y(t_j; y0) by −M(t_j)A^{-2}y0. Every subsequent implication in Theorem 1.7 and Theorem 1.9 inherits this replacement. The additional gap in Proposition 3.1, Step 1, concerning multiplication by χ_{ω_j} after H^{-6} convergence, is real but repairable via the continuous map Φ(t_j): H^{-4} → L2 and weak L2 convergence, so it does not change the overall risk profile. Because the unresolved lemma is outside what this preprint actually proves, the appropriate verdict remains CONDITIONAL rather than ACCEPT; it is not a demonstrated contradiction, so REJECT is too strong. The reader's assessment already reflects this, so no adjustment is needed. I agree with the reader's verdict and weakest-assumption identification.","tokens_in":15278,"tokens_out":5050,"duration_ms":49085,"concrete_test":"Fix a non-analytic C^2 kernel, for instance M(t) = e^{-1/t^2} for t > 0 and M(0) = 0, or a compactly supported C^2 bump. For the scalar Volterra equation (5.2), compute x_k(t) numerically for λ_k = 10, 100, 1000 at a fixed t ∈ (0,1) and check whether |x_k(t) + M(t)/λ_k^2| ≤ C λ_k^{-3} with C independent of λ_k; equivalently, test whether R(t, λ_k) := λ_k^3(x_k(t) + M(t)/λ_k^2) stays bounded. Independently, re-derive (2.3) and (2.4) from the proofs of [16, Theorems 1.1–1.2] using only M ∈ C^2, tracking every place where analyticity of M is otherwise invoked. If the numerical remainder grows faster than λ_k^{-3}, or if the re-derivation requires analyticity at some step, Lemma 2.1 is false or unproved, invalidating the central results.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing premise is Lemma 2.1, equation (2.3): Φ(t) = −M(t)A^{-2} + R(t,A)(tA)^{-3}, with R(t,A) bounded on H^s for every s and satisfying the smoothing estimate (2.4). This decomposition is used in every main step: Proposition 2.3 replaces y(t_j; y0) by −M(t_j)A^{-2}y0 plus a smoother remainder, Proposition 3.1 uses the same replacement to show AV ⊂ V, and Theorem 1.7 uses both. The paper quotes the lemma from [17, Corollary 2.2], where the kernel was assumed analytic, and Remark 2.2 asserts, without proof, that it still holds for M ∈ C^2([0,∞)). The phrase 'after carefully checking its proof' is not a proof accessible to the reader, and [17] is listed only as 'to appear.' Since the sharp characterization of observability rests on this exact asymptotic, an unverified extension to C^2 is a genuine gap. A secondary, repairable gap occurs in Proposition 3.1, Step 1: strong convergence in H^{-6} does not by itself justify passing χ_{ω_j}y(t_j; y_{0,k}) = 0 to the limit, because multiplication by a characteristic function is not bounded on H^{-6}; this can be fixed by using Φ(t_j) ∈ L(H^{-4}, L2) and weak L2 convergence. The central unresolved risk remains Lemma 2.1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies finite-time-sampling observability for the heat equation with memory ∂ty − ∆y + ∫0^t M(t−s)y(s)ds = 0 on a bounded domain, with observations made at finitely many instants tj on small open sets ωj. The main results are a two-sided sampling observability inequality (1.2), a sharp sufficient geometric condition (1.4) under a backward-uniqueness assumption (Theorem 1.7), and an equivalence between the existence of some observation sets and backward uniqueness plus the condition ∑_j |M(tj)| > 0 (Theorem 1.9). The proofs rest on an asymptotic expansion of the solution semigroup (Lemma 2.1), a relaxed observability inequality (Proposition 2.3), and a unique continuation property at finitely many instants (Proposition 3.1). Section 5 translates backward uniqueness into conditions on the zero sets N_k of scalar Volterra solutions and gives examples for kernels M(t) ≤ 0, M(t)=t, and M(t)=ce^{αt}.","tokens_in":15518,"tokens_out":8814,"duration_ms":88697,"significance":"If the main results are correct, this is the first treatment of impulsive, finite-time sampling observability for heat equations with memory, and it provides a clean geometric characterization: under backward uniqueness, observability is governed by the pointwise covering condition ∑_j |M(tj)| χ_{ωj}(x)>0. The paper also gives a constructive procedure for choosing time instants and observation regions from the kernel, together with concrete examples. The main theorems are novel relative to the existing interval-observation literature for memory heat equations. The proofs are self-contained except for the central semigroup expansion, which is imported from the authors' prior work [17] (listed as 'to appear') and extended from analytic to C^2 kernels without a supplied proof; because every main statement relies on this expansion, the current version is not yet fully verifiable.","major_comments":[{"comment":"The asymptotic expansion (2.3), Φ(t)=−M(t)A^{−2}+R(t,A)(tA)^{−3}, is the load-bearing tool: it is used in Proposition 2.3, Proposition 3.1, Theorem 1.7, and Theorem 1.9. However, Lemma 2.1 is quoted from [17, Corollary 2.2], and Remark 2.2 states that [17] assumed an analytic kernel and that the result 'still holds' for M∈C^2([0,∞)) 'after carefully checking its proof.' No proof, or even a sketch of the verification, is given, and [17] is listed as 'to appear.' Since all subsequent results inherit this expansion, the C^2 extension is an unverified hypothesis. Please either provide a complete proof of (2.3) under assumption (A), or restrict the main theorems to analytic kernels (with a separate argument for which parts of Section 5 survive).","section":"Section 2, Lemma 2.1 and Remark 2.2"},{"comment":"After obtaining Φ(tj)y0,k→Φ(tj)ŷ strongly in H^{−6} (equation (3.6)), the proof concludes from χ_{ω_j}Φ(tj)y0,k=0 that χ_{ω_j}Φ(tj)ŷ=0. This inference is not valid as written, because multiplication by a characteristic function is not a bounded operator on H^{−6}. The gap is repairable: by Lemma 2.1, Φ(tj) maps H^{−4} continuously into L^2(Ω), and the weak convergence y0,k⇀ŷ in H^{−4} implies χ_{ω_j}Φ(tj)y0,k⇀χ_{ω_j}Φ(tj)ŷ weakly in L^2(Ω); hence the limit is zero. Please revise this step accordingly.","section":"Section 3, Proposition 3.1, Step 1"}],"minor_comments":[{"comment":"Before applying Definition 1.3 to the eigenfunction e, the proof asserts e∈L^2(Ω) without justification. This can be derived from Step 2: since AV⊂V, one has A^2e∈V⊂H^{−4}, so e=A^{−2}(A^2e)∈L^2(Ω). Please state this one-line argument explicitly.","section":"Section 3, Step 3"},{"comment":"For x0∈∂Ω, the normalized function |B(x0,1/k)|^{−1/2}χ_{B(x0,1/k)∩Ω} has L^2 norm tending to 2^{−1/2} rather than 1, so the equality in (2.10) is not correct for boundary points. Since (1.5) is an almost-everywhere condition, the proof may simply restrict the arbitrary point x0 to the open set Ω; please make this restriction explicit.","section":"Proposition 2.3, equations (2.9)–(2.10)"},{"comment":"The construction of {~ω}_j=1^m from {ωhat_j}_{j∈J} is imprecise: if J is a proper subset of {1,...,m}, the phrase 'subsequence' does not specify what ~ω_j should be for j∉J. Please define ~ω_j for these indices explicitly (for instance, as arbitrary nonempty open sets, since the corresponding terms vanish in (4.9)).","section":"Theorem 1.9, Step 2"},{"comment":"In (5.10), the constants α_k, β_k, φ_k and C̃_k are not defined; since the subsequent countability claim depends only on α_k<0, please state their defining relations or move the computation to a remark.","section":"Example 5.9"},{"comment":"There are minor typographical issues, including 'e quations' in the abstract and inconsistent use of R+ for (0,+∞) versus [0,+∞); please proofread and make the notation uniform.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main unresolved risk is the C^2 extension of Lemma 2.1 from the analytic case in [17]. If the authors cannot supply the proof, the main theorems should be restated under analytic kernels. The paper also leans heavily on [16,17], which are self-citations, and [17] is 'to appear'; this makes the verification harder for a referee and for readers. The Proposition 3.1 Step 1 issue is local and fixable, and the other comments are minor."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my take. The paper is the first to study observability for heat equations with memory when observations are made at a finite set of time instants, and it delivers a sharp characterization: under a backward uniqueness assumption, the two-sided sampling inequality (1.2) is equivalent to a simple geometric covering condition involving the kernel values |M(t_j)|. That is a real contribution, not a repackaging of known results. The proof structure is coherent: Proposition 2.3 gives a clean relaxed observability inequality whose necessary and sufficient condition is the weak geometric condition, Proposition 3.1 provides the unique continuation step, and Theorems 1.7 and 1.9 follow. The backward uniqueness criteria in Section 5, especially the concrete kernel examples, are useful and checkable.\n\nThe paper is honest about its limitations, which I appreciate. It does not pretend to know how to go from the weak geometric condition to the full inequality, and it flags the gap in Remark 2.4.\n\nThe soft spots are real, though repairable. The load-bearing Lemma 2.1 is quoted from [17], which is listed as 'to appear' and is the authors' own work. In [17] the asymptotic expansion Φ(t) = −M(t)A^{−2} + R(t,A)(tA)^{−3} is proved for analytic kernels; Remark 2.2 asserts that it still holds for C^2 kernels 'after carefully checking its proof,' but no proof is supplied. Since every main step uses this expansion, the sharp characterization is only as solid as that unverified extension. A referee must see the proof or have access to [17]. The second issue is in Proposition 3.1, Step 1: the paper passes χ_{ω_j} Φ(t_j) y_{0,k} to the limit using strong convergence in H^{−6}, but multiplication by a characteristic function is not bounded on H^{−6}. That step needs repair, likely by using Φ(t_j) ∈ L(H^{−4}; L^2) and weak L^2 convergence. It is a local fix, not a structural flaw.\n\nOn balance, the central argument holds up if Lemma 2.1 is true. The paper deserves serious refereeing. I would not desk-reject it, and I would ask the referee to verify Lemma 2.1 carefully and to check that the C^2 extension is valid. If that check passes, this is a strong contribution to the control theory of memory PDEs.","headline":"First sharp sampled-observability result for heat equations with memory; worth refereeing, but the key asymptotic lemma is quoted from a to-appear paper and extended to C^2 kernels without proof.","tokens_in":16130,"tokens_out":2734,"would_cite":true,"duration_ms":24922,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B07","45K05","35K05","93C57"],"pacs":[],"model":"deepseek-v4-flash","headline":"A heat equation with memory is recoverable from finitely many instantaneous observations if and only if the kernel is nonzero at some sampling time and the chosen times avoid the zero sets of the scalar Volterra modes.","keywords":["heat equation with memory","sampling observability","two-sided observability inequality","geometric observation condition","backward uniqueness","impulse controllability","Volterra integro-differential equation","unique continuation"],"falsifier":"Take $\\Omega=(0,\\pi)$, $A$ the Dirichlet Laplacian, and a non-analytic $C^2$ kernel such as $M(t)=e^{-1/t}$ for $t>0$ and $M(0)=0$. Compute the scalar Volterra solutions $x_k(t)$ of (5.2) on the eigenbasis and test the consequence of (2.3), namely that $t^3\\lambda_k|\\lambda_k^2 x_k(t)+M(t)|$ stays bounded uniformly in $k$ for fixed $t>0$; a single sequence of modes violating this bound would disprove the lemma on which Theorems 1.7 and 1.9 depend.","tokens_in":14986,"feed_emoji":"📸","tokens_out":13220,"duration_ms":105316,"temperature":0.7,"pith_summary":"This paper tries to establish exactly when the initial state of a heat equation with memory can be reconstructed from finitely many instantaneous observations, each taken on a small open set at its own time. The main theorem says that, once a natural “backward uniqueness” condition holds at the chosen times, the two-sided sampling observability inequality is equivalent to a weighted geometric condition on the observation regions: the sum of the kernel magnitudes $|M(t_j)|$ times the indicator functions of the regions must be positive everywhere. A companion result makes the condition sharp by showing that observability for some choice of regions holds if and only if backward uniqueness holds and at least one $M(t_j)$ is nonzero. The paper also gives a concrete recipe for choosing the sampling times from the zero sets of scalar Volterra equations, and shows through examples that the usable times depend strongly on the kernel.","feed_headline":"Finite snapshots recover heat-with-memory states if kernel is nonzero","feed_subtitle":"Sharp condition: sampling times must avoid scalar-mode zeros; observation regions must cover the domain.","key_machinery":"The load-bearing object is the asymptotic expansion (2.3), $\\Phi(t)=-M(t)A^{-2}+R(t,A)(tA)^{-3}$, with $A$ the Dirichlet Laplacian and $\\Phi(t)$ the solution operator of (1.1). It says that at any positive time the solution is $-M(t)A^{-2}$ applied to the initial data plus a remainder that is three powers of $A$ smoother; this is what turns spatial observation of the sample into a weighted coverage condition on the sets $\\omega_j$. The second essential mechanism is the scalar reduction of backward uniqueness: writing the solution in the eigenbasis of $-A$, each coefficient obeys $x_k'(t)+\\lambda_k x_k(t)+\\int_0^t M(t-s)x_k(s)\\,ds=0$ with $x_k(0)=1$, and a nonzero datum is invisible at $\\{t_j\\}$ exactly when some mode $k$ satisfies $x_k(t_j)=0$ for every $j$. The zero sets $N_k=\\{t: x_k(t)=0\\}$ therefore dictate which sampling instants are admissible.","core_discovery":"The central claim, Theorem 1.7, is that for a $C^2$ memory kernel $M$, if the equation has backward uniqueness at times $t_1,\\dots,t_m$ — meaning no nonzero initial state vanishes at all those times — then the geometric observation condition $\\sum_{j=1}^m |M(t_j)| \\chi_{\\omega_j}(x)>0$ on $\\Omega$ implies the two-sided inequality $C^{-1}\\|y_0\\|_{H^{-4}} \\le \\sum_{j=1}^m \\|y(t_j;y_0)\\|_{L^2(\\omega_j)} \\le C\\|y_0\\|_{H^{-4}}$, and this inequality in turn implies the same positivity almost everywhere. Theorem 1.9 completes the equivalence: such an inequality holds for some open observation sets if and only if backward uniqueness holds at $\\{t_j\\}$ and $\\sum_{j=1}^m |M(t_j)|>0$. The proof rides on the asymptotic expansion $\\Phi(t)=-M(t)A^{-2}+R(t,A)(tA)^{-3}$, which makes the dominant part of each sample proportional to $M(t_j)A^{-2}y_0$; the remainder is smooth enough to be absorbed. Backward uniqueness is then reduced, by separation of variables, to the zero sets of scalar Volterra equations, which is what makes the choice of sampling times depend on the kernel.","pith_inferences":["This suggests that, if the $C^2$ extension of Lemma 2.1 is valid, the observability constant can be quantified in terms of the minimum of $\\sum_j |M(t_j)|\\chi_{\\omega_j}$ over $\\Omega$ and the stability of backward uniqueness, making the result directly usable for numerical reconstruction.","The zero-set criterion points to a general sampling strategy for convolution memory kernels: choose a pair of instants whose separation is shorter than the smallest gap between consecutive zeros of any scalar mode, which for kernels with periodic or almost-periodic scalar solutions guarantees backward uniqueness without detailed knowledge of the domain's eigenbasis.","One could test the optimal regularity $H^{-4}$ numerically by reconstructing initial data from noisy samples for $M(t)=c e^{\\alpha t}$ and measuring how the reconstruction error scales with noise as the target norm is varied."],"forward_implications":["Under the theorem's hypotheses, observing the solution on small open sets at finitely many instants recovers the initial datum in $H^{-4}$, and $H^{-4}$ is the optimal regularity scale for that recovery.","Any sampling instant with $M(t_j)=0$ contributes nothing to the observation; only instants with nonzero kernel value can be used, and their observation regions must jointly cover all of $\\Omega$ (or all but a null set for the necessary condition).","Theorem 1.9 provides a design rule: choose times so that $\\sum_j |M(t_j)|>0$ and no eigenmode is killed at every chosen time, then choose regions satisfying (1.4); this yields the two-sided inequality.","For kernels such as $M(t)=c e^{\\alpha t}$, any two instants closer than $\\pi/\\sqrt{c}$ give backward uniqueness, so two snapshots suffice.","By standard duality, the same inequality yields impulse controllability: finitely many interior controls injected at times $T-t_j$ can steer the memory-heat equation to any target in $H^4$ at time $T$."],"supporting_citations":[{"why":"Supplies Corollary 2.2, the asymptotic expansion (2.3) and the smoothing estimate (2.4) on which Propositions 2.3 and 3.1 and Theorem 1.7 rest.","marker":"[17]"},{"why":"Provides the flow-decomposition and propagation-of-singularities results, the boundedness of $\\Phi(t)$ on $H^s$, and Lemma 3.3 used for the scalar Volterra representation (5.4).","marker":"[16]"},{"why":"Gives the semigroup existence and uniqueness theorem used to define the solution map and the backward-uniqueness notion.","marker":"[12]"},{"why":"Shows that time-varying observation regions can restore controllability for heat equations with memory, motivating the sampling design in this paper.","marker":"[3]"},{"why":"Supplies the unique continuation property of eigenfunctions used at the end of Proposition 3.1 to force $e=0$.","marker":"[10]"},{"why":"Provides the finite-time impulse controllability and sampling observability framework for coupled heat equations that the present paper extends to memory kernels.","marker":"[14]"},{"why":"Establishes sampling observability for linear autonomous ODEs under irregular sampling, the finite-dimensional baseline the paper generalizes.","marker":"[18]"}],"fun_headline_variants":["Memory kernel sets sampling times for heat observability","Sharp condition for sampling heat-with-memory states","Finite samples recover heat-with-memory if kernel non-vanishing","Two-sided inequality for sampled heat-with-memory states","Observability for heat-with-memory: times set by kernel"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole characterization rests on Lemma 2.1's asymptotic expansion (2.3), which was proved for analytic memory kernels; Remark 2.2 asserts without proof that it still holds for any $C^2$ kernel, and every main theorem uses that extension.","fun_headline_variants_meta":{"raw":{"variants":["Memory kernel sets sampling times for heat observability","Sharp condition for sampling heat-with-memory states","Finite samples recover heat-with-memory if kernel non-vanishing","Two-sided inequality for sampled heat-with-memory states","Observability for heat-with-memory: times set by kernel"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000417,"raw_usage":{"total_tokens":2144,"prompt_tokens":931,"completion_tokens":1213,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":547,"completion_tokens_details":{"reasoning_tokens":1133}},"tokens_in":547,"tokens_out":1213,"duration_ms":10768,"temperature":1.0,"reasoning_tokens":1133,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:15:23.832958+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $\\Omega=(0,\\pi)$, $A$ the Dirichlet Laplacian, and a non-analytic $C^2$ kernel such as $M(t)=e^{-1/t}$ for $t>0$ and $M(0)=0$. Compute the scalar Volterra solutions $x_k(t)$ of (5.2) on the eigenbasis and test the consequence of (2.3), namely that $t^3\\lambda_k|\\lambda_k^2 x_k(t)+M(t)|$ stays bounded uniformly in $k$ for fixed $t>0$; a single sequence of modes violating this bound would disprove the lemma on which Theorems 1.7 and 1.9 depend.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Corollary 2.2, the asymptotic expansion (2.3) and the smoothing estimate (2.4) on which Propositions 2.3 and 3.1 and Theorem 1.7 rest."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the flow-decomposition and propagation-of-singularities results, the boundedness of $\\Phi(t)$ on $H^s$, and Lemma 3.3 used for the scalar Volterra representation (5.4)."},{"cited_title":"Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983","cited_arxiv_id":null,"evidence_quote":"Gives the semigroup existence and uniqueness theorem used to define the solution map and the backward-uniqueness notion."},{"cited_title":"Chaves-Silva, X","cited_arxiv_id":null,"evidence_quote":"Shows that time-varying observation regions can restore controllability for heat equations with memory, motivating the sampling design in this paper."},{"cited_title":"Lin, A uniqueness theorem for parabolic equations , Comm","cited_arxiv_id":null,"evidence_quote":"Supplies the unique continuation property of eigenfunctions used at the end of Proposition 3.1 to force $e=0$."},{"cited_title":"Qin and G","cited_arxiv_id":null,"evidence_quote":"Provides the finite-time impulse controllability and sampling observability framework for coupled heat equations that the present paper extends to memory kernels."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes sampling observability for linear autonomous ODEs under irregular sampling, the finite-dimensional baseline the paper generalizes."}],"review_version":1}