{"id":"6d7e7c77-66a1-4f11-bdb3-557729a175d5","arxiv_id":"2412.02046","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For damped nonlocal wave equations, equality of exterior measurements uniquely determines the damping coefficient and the potential (or nonlinearity).","lead":"This paper proves new uniqueness results for inverse problems involving damped, nonlocal wave equations, showing that boundary measurements can recover both the damping coefficient and the potential or nonlinearity. A new Runge approximation theorem and a theory of very weak solutions are the tools that make the simultaneous recovery possible.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.3 contains a logical gap: after deriving (γ1−γ2)∂t v1=0, the proof replaces ∂t v1 by ∂t(v1−η) in the integral identity without justification; a linearization argument (v1=v2) would repair the proof, but it is not given.","rationale":"The reader's conditional verdict is well-founded. I agree that the central new result, Theorem 1.3, is not proven as written because the inference from (γ1−γ2)∂t v1=0 to the integral identity with ∂t(v1−η) is invalid; the author substitutes a time-derivative of the exterior condition without justification. This is a genuine logical gap in the main semilinear theorem, not a matter of external consensus. I do not regard the cited multiplier estimate (2.23) as the decisive weakness: for γ∈C^{0,α}(R^n) with α>s, multiplication is bounded on \\tilde H^s(Ω) by a standard commutator estimate, and the paper's use of it in Theorem 2.8 and Lemma 3.2 is coherent. The proof of the linear theorem (Theorem 1.2) appears sound, as do the Runge approximation and the very weak solution framework; those parts have independent value. The semilinear gap is repairable by a short linearization argument that the author did not include, so the appropriate verdict is conditional acceptance rather than rejection. The paper should either supply the missing linearization step or otherwise prove the gamma-recovery identity from (γ1−γ2)∂t v1=0.","tokens_in":20488,"tokens_out":21362,"duration_ms":192442,"concrete_test":"Check the proof of Theorem 1.3: replace the passage after (3.21) with the linearization step (u_ε^{(1)}=u_ε^{(2)} and R_ε^{(j)}=O(ε^{r+1}) imply v1=v2, hence Theorem 1.2 yields γ1=γ2). If this replacement is valid, the theorem's conclusion stands. To confirm the flaw, evaluate the claimed identity with a concrete η that depends on t: the difference between the left side and the actual consequence of (γ1−γ2)∂t v1=0 equals −∫ (γ1−γ2)∂t η (w2−ψ)^⋆, which is generically nonzero.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The most load-bearing concern is in the proof of Theorem 1.3, Section 3.4. After the scaling limit in (3.19)–(3.21) the author obtains (γ1−γ2)∂t v1 = 0 in Ω_T. The next sentence claims this 'ensures' the integral identity ∫_{Ω_T} (γ1−γ2)∂t(v1−η)(w2−ψ)^⋆ dxdt = 0 for all ψ ∈ C_c^∞((W2)_T). This does not follow: ∂t(v1−η) = ∂t v1 − ∂t η, and ∂t η is generally nonzero, so the identity would imply −∫ (γ1−γ2)∂t η (w2−ψ)^⋆ = 0, which is not a consequence of (γ1−γ2)∂t v1=0. The subsequent gamma-recovery argument therefore lacks a valid premise. The gap is repairable: because u_ε^{(1)}=u_ε^{(2)} for all ε>0, (3.14) and (3.18) give ε(v1−v2)=R_ε^{(2)}−R_ε^{(1)}=O(ε^{r+1}), so dividing by ε and letting ε→0 yields v1=v2 for every η. Thus the linear DN maps for (γ1,0) and (γ2,0) agree, and Theorem 1.2 gives γ1=γ2. This repair is not present in the manuscript. The multiplier estimate (2.23) from [CRTZ24] is an external, likely correct lemma; it is not the decisive weakness.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a very weak solution theory for damped nonlocal wave equations L_γ u := ∂_t^2 u + γ∂_t u + (−Δ)^s u with Hölder-continuous damping γ, proves an optimal Runge approximation theorem in L^2(0,T;\\tilde H^s(Ω)), and applies it to Calderón-type inverse problems. The main theorems state that the partial Dirichlet-to-Neumann map simultaneously determines the pair (γ,q) in the linear case (Theorem 1.2) and the pair (γ,f) for r+1 homogeneous weak nonlinearities (Theorem 1.3).","tokens_in":20822,"tokens_out":6966,"duration_ms":64442,"significance":"The very weak solution theory for damped nonlocal wave equations with variable C^{0,α} damping (Theorem 2.8) and the resulting Runge approximation (Theorem 3.1) are genuine technical contributions that go beyond the undamped and constant-damping cases treated in earlier work. If the gap in the proof of Theorem 1.3 is repaired, the simultaneous determination results are natural but substantial extensions of [LTZ24a] and [Zim24]. The paper is clearly written and the main proofs are detailed, with no fitted parameters or numerical evidence; the arguments are analytic.","major_comments":[{"comment":"The inference from (γ1−γ2)∂_t v1 = 0 to the integral identity ∫_{Ω_T} (γ1−γ2)∂_t(v1−η)(w2−ψ)^⋆ dxdt = 0 is not justified. Since ∂_t(v1−η)=∂_t v1−∂_t η and η is only assumed to lie in C_c^∞((W1)_T), ∂_t η is generally nonzero, so the displayed identity is not a consequence of (γ1−γ2)∂_t v1=0. This is load-bearing because the subsequent gamma-recovery argument uses exactly this integral identity as the input to the Runge approximation and Lemma 3.2. The gap is repairable: because u_ε^{(1)}=u_ε^{(2)} for every ε>0, equations (3.14) and (3.18) give ε(v1−v2)=R_ε^{(2)}−R_ε^{(1)}=O(ε^{r+1}); dividing by ε and letting ε→0 yields v1=v2. Then the linear DN maps for (γ1,0) and (γ2,0) coincide, and Theorem 1.2 gives γ1=γ2. This repair should be inserted before the f-recovery step.","section":"Section 3.4, after Eq. (3.21)"},{"comment":"The proof of Theorem 2.8 uses the multiplier estimate (2.23), cited as [CRTZ24, Lemma 3.1], for multiplication by γ∈C^{0,α}(R^n) on \\tilde H^s(Ω) in a bounded Lipschitz domain. Please verify that the cited lemma applies verbatim with only Lipschitz boundary regularity; if it requires smoother boundary, the hypotheses of Theorems 2.8, 3.1, 1.2 and 1.3 must be adjusted accordingly. This is a correctness-risk point rather than an observed contradiction, but it should be checked explicitly.","section":"Section 2.2, Theorem 2.8"}],"minor_comments":[{"comment":"The bibliography lists [LTZ24b] and [LTZ24c] with the identical title and journal data; this appears to be a duplicate reference and should be corrected.","section":"References"},{"comment":"In the displayed computation in Lemma 3.2, the phrase “in the second equality we used (3.4), Ψ(T)=0 and (3.4)” contains a duplicated citation; the intended condition should be stated clearly.","section":"Proof of Lemma 3.2"},{"comment":"The final recovery of f1=f2 is delegated to “[LTZ24a, p. 29]” without explaining why the argument carries over to the damped equation with γ present; a short justification or a precise statement of the needed lemma would make the proof more self-contained.","section":"Section 3.4, final step"},{"comment":"In the proof of (2.26), the regularization v_ε is introduced via a parabolic regularization, but the displayed convergence in (2.27) does not explicitly state the convergence of ∂_t v_ε needed to pass the term ⟨u, γ∂_t v_ε⟩; please add the missing convergence statement or a reference for it.","section":"Section 2.2, Eq. (2.26)"}],"recommendation":"major_revision","confidential_remarks":"The paper leans heavily on the author's prior works for the proof architecture and final nonlinear-recovery argument, but the damped very weak solution theory and the Runge approximation are genuinely new. The main issue is the gap in Theorem 1.3, which is repairable by the linearization argument described in the major comments; after that repair the contribution should be publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere's my read on Zimmermann's arXiv:2412.02046. The new material is real: an optimal Runge approximation for damped nonlocal wave equations with variable damping coefficient γ, a very weak solution theory for L_γ, and simultaneous determination of γ with either a linear potential q or a homogeneous nonlinearity f. The linear theorem (1.2) is proved convincingly, and the Runge theorem (3.1) extends the earlier framework from [LTZ24a] in the natural way. The damped-operator results (Theorems 2.7, 2.8) are genuinely new and do not reduce to the γ=0 cases; reliance on the author's own prior work is substantial but not circular.\n\nThe real soft spot is in the proof of Theorem 1.3. After scaling, the author derives (γ1−γ2)∂t v1 = 0 in Ω_T. The next step replaces ∂t v1 by ∂t(v1−η) in the integral identity, which does not follow without ∂t η = 0. That is a genuine logical gap, and the reader and stress-test note agree on this. The repair is straightforward: since u_ε^(1)=u_ε^(2) for all ε, equations (3.14) and (3.18) give ε(v1−v2)=O(ε^{r+1}), so v1=v2 after dividing by ε; then the linear DN maps agree and Theorem 1.2 gives γ1=γ2. But this argument is not in the manuscript. As written, the gamma recovery step in the semilinear theorem lacks a valid premise.\n\nI don't see other load-bearing issues. The multiplier estimate (2.23) from [CRTZ24] is external, but Lemma 3.1 there covers C^{0,α} multipliers on H~s(Ω) for bounded Lipschitz domains; it's not the weak point. The writing is clear, and the very weak solution theory is carefully done.\n\nWho is this for? People working on fractional Calderón problems for hyperbolic equations. It's a solid incremental step, not a paradigm shift. It deserves a serious referee: the linear result is clean, and the semilinear gap is specific and repairable. I'd recommend sending it to review with a request to fix the gamma recovery step in Theorem 1.3. Conditional acceptance.","headline":"Solid extension of the fractional Calderón program to variable damping, with a specific and repairable gap in the semilinear theorem.","tokens_in":21425,"tokens_out":1761,"would_cite":true,"duration_ms":15049,"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":"For the damped nonlocal wave equation, exterior boundary measurements uniquely determine both the damping coefficient and the perturbation.","keywords":["fractional Laplacian","damped wave equation","Runge approximation","very weak solutions","inverse problem","Dirichlet-to-Neumann map","semilinear wave equation","simultaneous determination"],"falsifier":"One concrete check is to test estimate (2.23) on a bounded Lipschitz domain with a corner, such as a polygon, with $\\gamma(x)=|x_1|^\\alpha$ and $v\\in\\tilde H^s(\\Omega)$; if the quotient $\\|\\gamma v\\|_{\\tilde H^s(\\Omega)}/\\|v\\|_{\\tilde H^s(\\Omega)}$ were not bounded by a constant times $\\|\\gamma\\|_{C^{0,\\alpha}}$, the fixed-point proof of Theorem 2.8 would collapse. A second test is numerical: compute the partial DN map for two pairs $(\\gamma_1,q_1)$ and $(\\gamma_2,q_2)$ with $\\gamma_1\\ne\\gamma_2$ or $q_1\\ne q_2$ on a simple domain; the theorem asserts these maps can never agree on the chosen exterior sets.","tokens_in":20193,"feed_emoji":"🌊","tokens_out":9726,"duration_ms":288168,"temperature":0.7,"pith_summary":"The paper establishes a simultaneous uniqueness result for boundary inverse problems for damped nonlocal wave equations of the form $L_\\gamma u + f(u)=0$ with $L_\\gamma=\\partial_t^2+\\gamma\\partial_t+(-\\Delta)^s$. It proves that the partial exterior-to-boundary DN map determines both the Hölder-continuous damping coefficient $\\gamma$ and the equation's perturbation: either a linear potential $q\\in L^p(\\Omega)$ or a homogeneous weak nonlinearity $f$. The key step is an optimal Runge approximation theorem showing that solutions generated by exterior data are dense in the energy space $L^2(0,T;\\tilde H^s(\\Omega))$, which lets equality of boundary measurements be converted into integral identities that force coefficient equality inside the domain. This matters because it removes the damping term from being an obstruction in nonlocal wave inverse problems and recovers two coefficients at once from partial exterior data.","feed_headline":"Exterior wave data fix damping and source term at once","feed_subtitle":"A Runge approximation theorem makes the boundary map recover both coefficients for fractional wave equations.","key_machinery":"The carrying object is the very weak solution theory for the damped operator $L_\\gamma=\\partial_t^2+\\gamma\\partial_t+(-\\Delta)^s$, defined by duality against solutions of the time-reversed adjoint problem. Well-posedness is obtained by rewriting the equation as $(\\partial_t^2+(-\\Delta)^s)u=F-\\gamma\\partial_t u-qu$ and applying a Banach fixed point argument in $C([0,T];\\tilde L^2(\\Omega)\\times H^{-s}(\\Omega))$; the Hölder regularity of $\\gamma$ enters through the multiplier estimate $\\|\\gamma v\\|_{\\tilde H^s(\\Omega)}\\le C\\|\\gamma\\|_{C^{0,\\alpha}(\\mathbb{R}^n)}\\|v\\|_{\\tilde H^s(\\Omega)}$ on Lipschitz domains. This theory yields the Runge approximation theorem: the set $\\{u_\\varphi-\\varphi:\\varphi\\in C_c^\\infty(W_T)\\}$ is dense in $L^2(0,T;\\tilde H^s(\\Omega))$, proved by a Hahn–Banach argument that reduces density to the unique continuation property of the fractional Laplacian. The recovery step then combines the integral identity with controlled limit passages: Lemma 3.2 handles the time-derivative term in the linear case, and an $\\varepsilon$-scaling argument isolates the damping coefficient before the nonlinearity in the semilinear case.","core_discovery":"The central claim is Theorems 1.2 and 1.3: for a bounded Lipschitz domain $\\Omega$, $0<s<\\alpha\\le 1$, and coefficient pairs $(\\gamma_j,q_j)\\in C^{0,\\alpha}(\\mathbb{R}^n)\\times L^p(\\Omega)$ with $p$ in the stated range, equality of the partial boundary-to-boundary maps $\\Lambda_{\\gamma_1,q_1}$ and $\\Lambda_{\\gamma_2,q_2}$ on $W_1\\times W_2$ implies $\\gamma_1=\\gamma_2$ and $q_1=q_2$ in $\\Omega$. The same conclusion holds for $r+1$ homogeneous weak nonlinearities $f_1,f_2$ with $r>0$, yielding $f_1=f_2$ on $\\Omega\\times\\mathbb{R}$. In other words, the paper shows that damping and perturbations of fractional wave equations are determined simultaneously by exterior measurements, and that the previously known $\\gamma=0$ results extend to variable Hölder-continuous damping.","pith_inferences":["The paper does not develop this, but the same fixed-point and multiplier framework is a natural route to time-dependent damping $\\gamma(t,x)$, provided the multiplier estimate holds uniformly in time.","The $\\varepsilon$-scaling recovery of the nonlinearity suggests a quantitative version: the rate at which $f_1(u_\\varepsilon)-f_2(u_\\varepsilon)$ vanishes as $\\varepsilon\\to0$ could yield a stability estimate for $f$, though no stability bound is proved here.","Because the Runge density is in $L^2(0,T;\\tilde H^s(\\Omega))$ rather than $L^2(\\Omega_T)$, the method may also apply to recovery of lower-order terms such as drift or conductivity in damped nonlocal wave equations, a direction the paper leaves open.","A numerical experiment testing the Runge approximation for variable $\\gamma$ on a polygonal domain would give a practical check of the multiplier estimate's sharpness; the paper contains no numerics."],"forward_implications":["For the linear model, equality of the partial DN maps implies $\\gamma_1=\\gamma_2$ and $q_1=q_2$ inside $\\Omega$, so damping and potential are recovered together rather than sequentially.","For semilinear models with $r+1$ homogeneous weak nonlinearities, the same measurements determine the damping coefficient and the whole function $f$ on $\\Omega\\times\\mathbb{R}$, including growth exponents $r>1$ that earlier methods could not reach.","The Runge approximation is optimal in the sense that the approximation space is the natural energy space $L^2(0,T;\\tilde H^s(\\Omega))$, allowing recovery of $q\\in L^p(\\Omega)$ at the lowest regularity the estimates permit.","All results hold with partial exterior data: $W_1$ and $W_2$ are arbitrary nonempty open subsets of the exterior $\\Omega^e$, so full-boundary measurements are not needed.","The very weak solution theory covers inhomogeneous sources and nonzero initial data, so the identification argument applies to a wider class of boundary-to-boundary measurements than those used in the main theorems."],"supporting_citations":[{"why":"Supplies Lemma 3.1, the multiplier estimate (2.23) for Hölder functions on Lipschitz domains, which carries the variable damping in the fixed-point proof.","marker":"[CRTZ24]"},{"why":"Provides the very weak solution framework for gamma equal to zero together with the optimal Runge approximation template and the polyhomogeneous recovery strategy used at the end of Theorem 1.3.","marker":"[LTZ24a]"},{"why":"Supplies well-posedness, energy estimates, and the multiplication bound for qu used throughout the weak and very weak solution sections.","marker":"[LTZ24c]"},{"why":"Establishes the unique continuation principle for the fractional Laplacian, which turns exterior vanishing into global vanishing in the Runge approximation proof.","marker":"[GSU20]"},{"why":"Supplies the abstract second-order evolution existence theory and energy identities used in Theorem 2.1 and in the parabolic regularization inside Theorem 2.8.","marker":"[DL92]"},{"why":"Supplies the viscous wave equation setting that this paper extends, together with the lemma controlling nonlinear terms in the epsilon-to-zero limit.","marker":"[Zim24]"}],"fun_headline_variants":["Damping and source term fixed by exterior wave data","Nonlocal wave boundary map reveals damping and source","Optimal Runge approximation enables simultaneous recovery","Exterior data determine damping and nonlinearity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that multiplication by a $C^{0,\\alpha}(\\mathbb{R}^n)$ function is a bounded map on the fractional Sobolev space $\\tilde H^s(\\Omega)$ for bounded Lipschitz domains, as used in estimate (2.23). If Hölder-continuous damping required more boundary regularity than Lipschitz, the very weak solution theory and the Runge approximation for variable $\\gamma$ would not carry through.","fun_headline_variants_meta":{"raw":{"variants":["Damping and source term fixed by exterior wave data","Nonlocal wave boundary map reveals damping and source","Optimal Runge approximation enables simultaneous recovery","Exterior data determine damping and nonlinearity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000554,"raw_usage":{"total_tokens":2570,"prompt_tokens":808,"completion_tokens":1762,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":424,"completion_tokens_details":{"reasoning_tokens":1713}},"tokens_in":424,"tokens_out":1762,"duration_ms":13470,"temperature":1.0,"reasoning_tokens":1713,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:54:50.744891+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete check is to test estimate (2.23) on a bounded Lipschitz domain with a corner, such as a polygon, with $\\gamma(x)=|x_1|^\\alpha$ and $v\\in\\tilde H^s(\\Omega)$; if the quotient $\\|\\gamma v\\|_{\\tilde H^s(\\Omega)}/\\|v\\|_{\\tilde H^s(\\Omega)}$ were not bounded by a constant times $\\|\\gamma\\|_{C^{0,\\alpha}}$, the fixed-point proof of Theorem 2.8 would collapse. A second test is numerical: compute the partial DN map for two pairs $(\\gamma_1,q_1)$ and $(\\gamma_2,q_2)$ with $\\gamma_1\\ne\\gamma_2$ or $q_1\\ne q_2$ on a simple domain; the theorem asserts these maps can never agree on the chosen exterior sets.","supporting_citations":[],"review_version":1}