{"id":"531e6a54-82fd-4782-a599-67ff9b83b256","arxiv_id":"2509.04850","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims unique coefficient recovery for a parabolic-elliptic-elliptic chemotaxis system from boundary data, but the proof relies on invalid arbitrary initial data for the elliptic equations.","lead":"This paper claims that measuring the boundary and final-time values of non-negative solutions uniquely determines all coefficients in a nonlinear attraction-repulsion chemotaxis model with mixed parabolic-elliptic-elliptic equations. It would be the first identifiability result for such a system, but the proof contains load-bearing gaps in how it treats the elliptic components.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Elliptic components admit no free positive initial data, so the recovery of α01/β01 and all later steps has no valid input to vary; the proof also linearizes at zero despite claiming expansion at arbitrary constant equilibria.","rationale":"The reader's central objection is correct and is the single most load-bearing weakness. In the parabolic-elliptic-elliptic system (1.9)/(2.5), v and w are elliptic at each time, so their initial values are not independent degrees of freedom. The proof's recovery of α01 and β01 relies on choosing arbitrary positive g1,h1, deriving a homogeneous elliptic equation for v(I),w(I), and invoking a maximum principle for positivity. That step has no basis: arbitrary positive g1 is generally incompatible with the elliptic constraint, and positivity of the elliptic solution does not follow from positivity of a prescribed initial value. Since the recovery of α01 and β01 occurs at the first identification step and every subsequent linearization reuses v(I),w(I), the whole induction collapses. The additional inconsistency between the claimed expansion around arbitrary constant equilibria and the actual linearization around (0,0,0) is independently damaging: for the logistic nonlinearity F=ru−μu² at a positive equilibrium, the linearized equation in (2.7) has the wrong coefficient, so even the recovery of r is not justified away from the zero state. These are not stylistic or minor technical issues; they are failures of the assumptions needed for the proof's first moves. No machine-checked verification, reproducible code, or other independent support is present. The auxiliary Lemma 2.2 may be useful in other contexts, and the fully parabolic version may be treatable by the cited prior work, but the stated τ=0 theorem and Corollary 1.6 are not supported by the argument. The verdict of REJECT, with the possibility that a repaired proof under compatible elliptic data and a correctly derived linearization could establish a restricted version, is appropriate.","tokens_in":21558,"tokens_out":7054,"duration_ms":64295,"concrete_test":"Set τ=0, G=αu−βv, H=γu−δw in (2.5), choose f1=0 and g1≡1>0, h1≡1>0, and insert t=0 into the linearized v- and w-equations (2.7). The v-equation forces −α01=0 and the w-equation forces −β01=0; since α01=−β and β01=−δ in this model, the system has no solution for β,δ≠0. More generally, check whether any positive g1 satisfying the Neumann elliptic constraint 0=Δg1+G(x,0,g1) can still range over a full Fourier basis in the transverse variable; if not, the inversion over ξ′ used in (2.20) cannot be performed. Separately, re-derive the first-order u-equation around u0=r/μ>0 and compare with ∂tu(I)=Δu(I)+r u(I) in (2.7); the coefficient is r−2μu0=−r, not r, confirming the proof only covers zero equilibrium.","verdict_should_be":"REJECT","load_bearing_attack":"The weakest point is the treatment of the elliptic components in the τ=0 case. In (2.5)/(2.7) the v and w equations are elliptic at every time, so their values at t=0 are not freely prescribable: they must satisfy the elliptic compatibility 0=Δg+G(x,f,g) and 0=Δh+H(x,f,h). The proof nevertheless takes arbitrary positive g1,h1 in the first-order system (2.7) and uses the maximum principle to claim v(I),w(I)>0. This fails for two reasons. First, for a generic positive g1 no solution exists: with f1=0 and G=αu−βv, taking g1≡1 forces β=0. Second, with u(I)=0 the homogeneous elliptic equation for v(I) is −Δv(I)−α01v(I)=0 with Neumann boundary conditions, so its solutions are eigenfunctions of −Δ with eigenvalue α01; positivity of g1 does not imply that such a solution exists or is positive. The recovery of α01 in (2.18)–(2.20) and β01 in (2.27)–(2.29) depends on freely varying g1,h1 and on the positivity conclusion; once these inputs are unavailable, the Fourier-inversion step has no valid test functions, and the later second- and higher-order recoveries that reuse v(I),w(I) inherit the gap. A compounding inconsistency is that the proof linearizes at (0,0,0), as stated in Section 2.2 ('it is easy to check that ... (u_j,v_j,w_j) = (0,0,0) is the solution ... when ε=0'), while Definitions 1.1–1.4 and Theorem 1.5 assert expansion around an arbitrary non-negative constant solution. For F=ru−μu² with u0=r/μ>0, the first derivative is r−2μu0=−r, not r, so (2.7) is even the wrong linearization at nonzero equilibria.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the inverse problem of determining the coefficients χ, ξ, and the Taylor coefficients of the nonlinearities F, G, H in a parabolic-elliptic-elliptic chemotaxis system from boundary traces and final-time measurements of non-negative solutions. The main result, Theorem 1.5, asserts that equality of the measurement maps M_A1^+ and M_A2^+ over all admissible initial data implies A_1 = A_2 in Ω × R. The proof is based on high-order linearization around a constant solution, construction of CGO solutions, Fourier inversion, and a multiplicative-separability uniqueness lemma, with a final application to an attraction-repulsion chemotaxis model with logistic growth in Corollary 1.6.","tokens_in":22043,"tokens_out":6811,"duration_ms":60606,"significance":"If Theorem 1.5 were valid, it would be a substantial contribution: unique recovery of all coefficients and all Taylor coefficients in a mixed parabolic-elliptic-elliptic system from boundary data appears to be new and would extend the authors' earlier parabolic-system results. The paper is clearly organized, states explicit admissible classes, and attempts to handle the nonlinear structure through high-order variations. However, several load-bearing technical assumptions in the τ = 0 proof are not justified, and at least one is inconsistent with the stated generality of the theorem. The central claim is therefore not established by the arguments presented.","major_comments":[{"comment":"In the τ = 0 case, v and w satisfy elliptic equations at every time, so their initial values g and h cannot be prescribed independently of f. The first-order system (2.7) nevertheless imposes both an elliptic equation and an initial condition on v^(I) and w^(I); for a generic positive g1 no solution exists. For instance, with f1 = 0, g1 ≡ 1, and G = αu − βv, the elliptic compatibility condition at t = 0 reduces to β = 0. Consequently, the variation of g1 and h1 used to recover α01 in (2.18)-(2.20) and β01 in (2.27)-(2.29) has no admissible test functions, and the conclusions α01_1 = α01_2 and β01_1 = β01_2 are unsupported. All later steps that reuse v^(I) and w^(I) inherit this gap.","section":"§2.2, Eqs. (2.5)-(2.7)"},{"comment":"The proof states that '(u_j, v_j, w_j) = (0,0,0) is the solution ... when ε = 0', but Definitions 1.1-1.4 and Theorem 1.5 assert expansion around an arbitrary known non-negative constant solution (u0,v0,w0). Linearizing at (0,0,0) gives the first-order u-equation in (2.7), whereas the linearization around a general constant state would contain the terms −χ_j u0 Δv^(I) + ξ_j u0 Δw^(I) and the coefficient r_j − 2μ_j u0. These omitted terms and the incorrect coefficient are load-bearing for the recovery of r, χ, ξ, and μ in §2.2-§2.3. In particular, for the logistic nonlinearity the only constant solutions are u0 = 0 and u0 = r/μ, so the claimed generality around arbitrary constant solutions is not achieved.","section":"§2.2, linearization point"},{"comment":"The device 'By controlling the initial data so that Δv^(I)(x,t) = Δw^(I)(x,t) = 0' is not available for the elliptic components. The elliptic equation in (2.7) determines Δv^(I) as −α10(x)u^(I) − α01 v^(I), and v^(I) itself is not freely prescribable; imposing Δv^(I) = 0 forces v^(I) = −(α10/α01)u^(I) up to a harmonic function, which is generally incompatible with the Neumann boundary condition and with the independent positive choices of initial data used elsewhere. The subsequent derivation of (2.35)-(2.36) and the recovery of χ, ξ, and μ in (2.38)-(2.46) therefore lack a valid input-data construction.","section":"§2.3, Eqs. (2.34)-(2.36)"},{"comment":"The proof of recovery of all higher-order coefficients is not carried out. After the second-order step, §2.4 states only that 'The main idea ... is mathematical induction, based on the k-th variation' and then declares the proof complete. No induction hypothesis, no compatibility conditions for the elliptic components at order k, and no construction of admissible test data are given. Since Theorem 1.5 asserts equality of the full Taylor coefficients in Ω × R, this is an essential missing part of the proof, not a mere presentation detail.","section":"§2.4, higher-order recovery"},{"comment":"Lemma 2.2, used in §2.3 to recover α02, α11, β02, and related coefficients, is proved by separating variables as if Ω were a rectangular product Ω' × (a,b). For a general bounded Lipschitz domain the cross-section {x_n : (x',x_n) ∈ Ω} depends on x', so the factorization in (2.2) and the assertion that the x'-integrals are Fourier transforms are not valid. The lemma may hold for product domains, but as stated it is not established for the domains permitted by Theorem 1.5.","section":"§2.1, Lemma 2.2"}],"minor_comments":[{"comment":"The condition 'ℓ ∈ N, N > 2' should read 'ℓ ∈ N, ℓ ≥ 3'.","section":"§2.4, line 1"},{"comment":"The citation markers such as '[1111–1313]' and '[ 11,22,55,66,2727]' appear to be formatting artifacts of the bibliography; they should be cleaned to standard reference numbers.","section":"Abstract and introduction"},{"comment":"The exponential factor e^{(2μ2+|ζ|^2−r)t} appears to be a typo; the exponent should involve the eigenvalue parameter θ of the heat-equation mode in (2.11), not μ2.","section":"Eq. (2.46)"},{"comment":"The quantifier 'For any (f,g,h) ∈ C^{2+α} × C^{2+α} × C^{2+α}' is too broad for the τ = 0 case because not every such triple is admissible for the elliptic equations; the statement should restrict to compatible initial data.","section":"§1.3, Theorem 1.5"}],"recommendation":"reject","confidential_remarks":"The core difficulty is not stylistic: the elliptic initial-value issue and the mismatch between the linearization point and the admissible classes invalidate the main theorem's proof. These are structural gaps that would require reformulating the problem and reworking the recovery of essentially all coefficients, so I do not see a route to a sound paper within the current scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nYou asked about arXiv:2509.04850. Quick take: it's a serious attempt at a genuinely open inverse problem, but the main proof does not hold together. The paper should not be accepted in its present form.\n\nWhat's new: the inverse problem for a parabolic-elliptic-elliptic chemotaxis system—recovering χ, ξ, and the coefficients of F, G, H from boundary and final-time observations of non-negative solutions—is new relative to the cited literature, which treats purely parabolic or elliptic systems. The high-order variation machinery is adapted from the authors' earlier work, but the application to the mixed-type system is a real extension. The writing is clear about the admissibility classes and the intended biological application (attraction-repulsion with logistic growth).\n\nThe soft spots are load-bearing. In the τ=0 case, v and w satisfy elliptic equations at each time. Their initial values cannot be freely prescribed, yet the proof does exactly that: the first-order system (2.7) includes v(I)(x,0)=g1(x), w(I)(x,0)=h1(x) with arbitrary positive g1,h1, and then uses a 'maximum principle for elliptic equations' to conclude v(I),w(I)>0. That is not valid: for a generic positive g1 no elliptic solution exists, and with u(I)=0 the homogeneous equation −Δv(I)−α01v(I)=0 has only eigenfunction solutions whose sign is not controlled by g1. The recovery of α01 and β01 depends on freely varying g1,h1 and on the positivity conclusion; without it the later Fourier-inversion steps have no valid test functions, and the higher-order recoveries that reuse v(I),w(I) collapse.\n\nCompounding this, the proof linearizes around (0,0,0). Section 2.2 states that (0,0,0) is the solution for ε=0, while Theorem 1.5 and Definitions 1.1–1.4 claim expansion around an arbitrary non-negative constant equilibrium. For F=ru−μu² with u0=r/μ>0, the first-order coefficient is r−2μu0=−r, not r, and the χu0Δv(I) term appears in the linearization—both absent from (2.7). So the proof does not match its own setup.\n\nMinor issues: differentiability of the solution map (2.6) is asserted rather than proved, and the recovery of higher-order coefficients is mostly an induction sketch. Also, Lemma 2.2 is borrowed and not fully proved, but that is fine if the source is solid.\n\nBottom line: the problem is worth solving and the paper points in a plausible direction, but the core argument is currently invalid. I would not cite it. A referee could usefully document these issues, and a repaired proof might work for the fully parabolic case with additional measurements, or for the elliptic case with compatible data. I'd send it to peer review with a clear expectation of major revision or rejection—but not simply desk-reject, because the question deserves expert attention.\n\nBest,","headline":"A genuinely new inverse problem and plausible strategy, but the proof's elliptic initial data and zero-state linearization are load-bearing flaws; reject as is.","tokens_in":22498,"tokens_out":4675,"would_cite":false,"duration_ms":37092,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","92-10","35Q92","35B09","35K99","35J99"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that boundary traces and final-time values of non-negative solutions of a parabolic-elliptic-elliptic chemotaxis system uniquely determine all model coefficients, including the attraction and repulsion sensitivities…","keywords":["parabolic-elliptic-elliptic system","chemotaxis","inverse problem","unique identifiability","boundary measurements","non-negative solutions","high-order variation","multiplicative separable form"],"falsifier":"For the $\\tau=0$ system (1.18), take positive initial data $(f,g,h)$ and solve the elliptic equations $0=\\Delta v+\\alpha u-\\beta v$ and $0=\\Delta w+\\gamma u-\\delta w$ at $t=0$; if the resulting $v(0),w(0)$ differ from the prescribed $g,h$, then the premise behind recovering $\\alpha_{01}$ and $\\beta_{01}$ fails. A decisive test is a numerical search over admissible configurations $A_1\\neq A_2$ for which the measurement maps $M^+_{A_1}$ and $M^+_{A_2}$ coincide on all non-negative admissible initial data; one such pair would refute Theorem 1.5, while a systematic absence of such pairs would support it.","tokens_in":21325,"feed_emoji":"🧫","tokens_out":11717,"duration_ms":93650,"temperature":0.7,"pith_summary":"This paper establishes a unique identifiability result for a coupled parabolic-elliptic-elliptic chemotaxis system: for two admissible parameter configurations, if the boundary traces and final-time values of their non-negative solutions agree for every admissible initial triple $(f,g,h)$, then the two configurations coincide in $\\Omega\\times\\mathbb{R}$. The recovered parameters include the chemotactic sensitivities $\\chi$ and $\\xi$, the logistic growth coefficients $r$ and $\\mu$, and every Taylor coefficient of the nonlinear chemical kinetic terms $G$ and $H$. A sympathetic reader would care because this is the first inverse-problem treatment of a mixed parabolic-elliptic-elliptic biological model, and it makes the measurement map a one-to-one readout of the model's parameters. The proof proceeds by high-order linearization around a constant solution, with each coefficient isolated by a suitable choice of initial data and converted into an integral identity tested against complex geometric optics solutions. A corollary applies the general theorem to an attraction-repulsion chemotaxis model with logistic growth, recovering the full parameter set $\\{\\chi,\\xi,r,\\mu,\\alpha,\\beta,\\gamma,\\delta\\}$.","feed_headline":"Boundary data fix every chemotaxis coefficient","feed_subtitle":"Attractant, repellent, growth, and reaction terms all recoverable from boundary and final-time data.","key_machinery":"The load-bearing machinery is high-order variation: initial data are expanded around a known constant solution $(u_0,v_0,w_0)$ as $f=u_0+\\varepsilon f_1+\\frac{\\varepsilon^2}{2}f_2+\\cdots$, and the $\\varepsilon^k$-derivatives of the solution satisfy linear parabolic-elliptic-elliptic systems. Each coefficient is recovered by choosing $f_1,g_1,h_1$ so that only the term of interest survives, then testing the resulting integral identity against special test functions. Two auxiliary tools carry the argument: Lemma 2.1, which provides parabolic test solutions of the form $u(x,t)=e^{\\theta t}l(x;\\theta)$ with $l$ a Neumann eigenfunction whose gradient does not vanish on any open set, and Lemma 2.2, a uniqueness result for multiplicative-separable functions $A(x)=A_1(x_1,\\dots,x_{n-1})A_2(x_n)$ that turns equal Fourier-type integrals into pointwise equality of the factors. Complex geometric optics solutions of the linearized elliptic and parabolic equations supply the oscillatory kernels $e^{\\zeta\\cdot x}$ and $e^{(|\\zeta|^2-r)t-i\\zeta\\cdot x}$, and the inverse Fourier transform together with the fundamental theorem of calculus converts vanishing time integrals into pointwise identities.","core_discovery":"The central claim is Theorem 1.5: under the paper's admissibility classes ($F\\in\\mathcal{A}$, $G\\in\\mathcal{B}$, $H\\in\\mathcal{C}$), with $\\chi,\\xi$ and all high-order variations independent of one spatial variable, equality of the measurement maps $M^+_{A_1}(f,g,h)=M^+_{A_2}(f,g,h)$ for all admissible non-negative initial data $(f,g,h)\\in C^{2+\\alpha}(\\Omega)^3$ implies $A_1=A_2$ in $\\Omega\\times\\mathbb{R}$. In words, the map from parameters to boundary and final-time observations is injective on the admissible set. The proof recovers the coefficients in layers: first-order variation gives $r$, $\\alpha_{01}$, $\\alpha_{10}(x)$, $\\beta_{01}$, $\\beta_{10}(x)$; second-order variation gives $\\chi$, $\\xi$, $\\mu$, and the quadratic Taylor coefficients $\\alpha_{11},\\alpha_{20},\\alpha_{02},\\beta_{11},\\beta_{20},\\beta_{02}$; induction on the order of variation gives all higher Taylor coefficients of $G$ and $H$. The same framework, specialized to the logistic attraction-repulsion model, yields Corollary 1.6: the entire biological parameter set $\\{\\chi,\\xi,r,\\mu,\\alpha,\\beta,\\gamma,\\delta\\}$ is uniquely determined by the measurement map, for both the parabolic-elliptic-elliptic ($\\tau=0$) and fully parabolic ($\\tau=1$) regimes.","pith_inferences":["Editorial inference: a natural next problem is stability; the paper establishes uniqueness but not quantitative stability, so the practical value for numerical reconstruction depends on an additional Lipschitz or logarithmic stability estimate that is not derived here.","Editorial inference: the admissibility conditions (independence of one spatial variable and multiplicative separability of higher Taylor coefficients) are used by the Fourier-transform argument of Lemma 2.2, so whether identifiability holds for fully general $x$-dependent coefficients is an open question the current proof does not settle.","Editorial inference: for the $\\tau=0$ case the inverse problem may need to be restated over the set of attainable initial data, since the elliptic equations for $v$ and $w$ determine their values from $u$ and boundary conditions; the proof's free choice of positive $g_1,h_1$ is the point where the mixed-type structure pushes back.","Editorial inference: the same high-order-variation-plus-CGO strategy looks transferable to other biological systems with elliptic components, such as chemotaxis models with nutrient or oxygen quasi-steady-state equations, provided the elliptic solution inherits the needed positivity and the initial values are compatible."],"forward_implications":["If Theorem 1.5 holds, the full parameter set of the parabolic-elliptic-elliptic chemotaxis model is identifiable from boundary and final-time observations, so unknown chemotaxis and kinetic coefficients can in principle be reconstructed without interior measurements.","The identifiability applies simultaneously to the $\\tau=0$ (mixed-type) and $\\tau=1$ (fully parabolic) regimes, so the result is stable across time-scale models of the same biological process.","Spatially dependent Taylor coefficients are recoverable, not just constants, under the multiplicative-separability condition, extending the inverse theory beyond constant-coefficient biological models.","Because only non-negative solutions are used, the identifiability statement is compatible with biologically meaningful population densities and avoids requiring sign-changing inputs.","The recovery of $r$, $\\chi$, $\\xi$, $\\mu$ by second-order variations, and all higher-order coefficients by induction, gives a complete parameter identifiability result rather than partial recovery of a few coefficients."],"supporting_citations":[{"why":"supplies the high-order variation method and the spectral test solutions of Lemma 2.1, credited with 'Proof. See [17].'","marker":"[17]"},{"why":"the prior inverse problem for a purely parabolic system with non-negative solutions whose admissible-class framework this paper generalizes and extends","marker":"[22]"},{"why":"the source of the multiplicative-separable uniqueness argument in Lemma 2.2, used to convert oscillatory integral identities into pointwise coefficient recovery","marker":"[23]"},{"why":"provides global existence for the fully parabolic attraction-repulsion system with proliferation, underpinning the tau=1 well-posedness","marker":"[9]"},{"why":"establishes well-posedness for the parabolic-elliptic-elliptic attraction-repulsion system, used for the tau=0 case","marker":"[29]"},{"why":"guarantees a unique uniformly bounded global classical solution for the logistic parabolic-elliptic-elliptic model underlying Corollary 1.6","marker":"[31]"}],"fun_headline_variants":["Boundary data uniquely identify all chemotaxis coefficients","Inverse chemotaxis: full parameter recovery from boundary","All chemotaxis terms recovered from boundary observations","Boundary measurement map pins down chemotaxis system","Complete coefficient identification for chemotaxis inverse problem"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that, in the $\\tau=0$ elliptic case, the initial values $g$ and $h$ of the chemical concentrations can be prescribed freely and that positivity of the linearized elliptic solutions $v^{(I)}$ and $w^{(I)}$ follows from positive $g_1$ and $h_1$; in a parabolic-elliptic-elliptic system those concentrations solve elliptic equations and are determined by $u$ and boundary conditions, so arbitrary positive initial values are generally not compatible and the positivity step is not guaranteed.","fun_headline_variants_meta":{"raw":{"variants":["Boundary data uniquely identify all chemotaxis coefficients","Inverse chemotaxis: full parameter recovery from boundary","All chemotaxis terms recovered from boundary observations","Boundary measurement map pins down chemotaxis system","Complete coefficient identification for chemotaxis inverse problem"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000326,"raw_usage":{"total_tokens":1906,"prompt_tokens":1105,"completion_tokens":801,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":721,"completion_tokens_details":{"reasoning_tokens":726}},"tokens_in":721,"tokens_out":801,"duration_ms":6994,"temperature":1.0,"reasoning_tokens":726,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:26:25.753664+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the $\\tau=0$ system (1.18), take positive initial data $(f,g,h)$ and solve the elliptic equations $0=\\Delta v+\\alpha u-\\beta v$ and $0=\\Delta w+\\gamma u-\\delta w$ at $t=0$; if the resulting $v(0),w(0)$ differ from the prescribed $g,h$, then the premise behind recovering $\\alpha_{01}$ and $\\beta_{01}$ fails. A decisive test is a numerical search over admissible configurations $A_1\\neq A_2$ for which the measurement maps $M^+_{A_1}$ and $M^+_{A_2}$ coincide on all non-negative admissible initial data; one such pair would refute Theorem 1.5, while a systematic absence of such pairs would support it.","supporting_citations":[{"cited_title":"On inverse problems in predator-prey models.Journal of Differential Equations, 397:349–376, 2024","cited_arxiv_id":null,"evidence_quote":"supplies the high-order variation method and the spectral test solutions of Lemma 2.1, credited with 'Proof. See [17].'"},{"cited_title":"Determining both sound speed and internal source in thermo-and photo-acoustic tomography.Inverse Problems, 31(10):105005, 2015","cited_arxiv_id":null,"evidence_quote":"the source of the multiplicative-separable uniqueness argument in Lemma 2.2, used to convert oscillatory integral identities into pointwise coefficient recovery"},{"cited_title":"Global existence in a fully parabolic attraction-repulsion chemotaxis system with singular sensitivities and proliferation","cited_arxiv_id":null,"evidence_quote":"provides global existence for the fully parabolic attraction-repulsion system with proliferation, underpinning the tau=1 well-posedness"},{"cited_title":"Well-posedness for a model derived from an attraction– repulsion chemotaxis system.Journal of Mathematical Analysis and Applications, 423(1):497–520, 2015","cited_arxiv_id":null,"evidence_quote":"establishes well-posedness for the parabolic-elliptic-elliptic attraction-repulsion system, used for the tau=0 case"},{"cited_title":"A parabolic–elliptic–elliptic attraction– repulsion chemotaxis system with logistic source.Journal of Mathematical Analysis and Applications, 455(1):650–679, 2017","cited_arxiv_id":null,"evidence_quote":"guarantees a unique uniformly bounded global classical solution for the logistic parabolic-elliptic-elliptic model underlying Corollary 1.6"}],"review_version":2}