{"id":"8062e913-dbc6-4237-8e3d-208bccbe005f","arxiv_id":"1908.04814","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Wave equations on compact curved spaces admit finite-dimensional global attractors when the damping acts only on an epsilon-controllable region whose interior and boundary measures are arbitrarily small.","lead":"This paper builds damping regions on curved spaces with boundary whose total size, including the part lying on the boundary, can be made arbitrarily small while still controlling every wave path. It then shows that wave equations with nonlinear damping and critical forcing on such regions have finite-dimensional long-term attractors.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.2(3) drops the boundary term ∫_{∂ω∩M} w ∂νw when differentiating ψ(t)=∫_ω w∂tw; unless this term is controlled, the quasi-stability inequality (4.18) and therefore Theorem 4.1 are not established.","rationale":"The central claim is the existence of a finite-dimensional smooth global attractor in Theorem 4.1, proved through gradient structure and quasi-stability. The most load-bearing internal step is not the quoted Carleman extension in Remark 3.1, though that is also asserted rather than proved; it is the boundary term dropped in Lemma 4.2(3). The quasi-stability estimate (4.18) is the only input to the finite fractal dimension and regularity conclusions, and it rests on the derivative estimate for ψ(t). Omitting ∫_{∂ω∩M}w∂νw is not a cosmetic issue: even for smooth ω, this trace pairing is generally nonzero and is not controlled by uniform L² estimates of ∂tw in ω. The concern is therefore concrete and checkable. In good faith, the gap appears repairable through a standard localized multiplier with a cutoff vanishing on ∂ω, and the geometric construction of ω is flexible enough that such a cutoff may be available. Thus I would not reject the paper or move the verdict from conditional; I recommend retaining the conditional verdict and requesting the boundary term be addressed before the proof is accepted.","tokens_in":20978,"tokens_out":11847,"duration_ms":132108,"concrete_test":"Recompute Lemma 4.2(3) retaining the boundary term I(t)=∫_{∂ω∩M} w∂νw. Check whether I(t) can be estimated using only the regularity w∈C_tH¹_0(M), ∂tw∈C_tL²(M), the equation (4.19), and assumptions (4.4)-(4.5), so that the final inequality (4.27) still holds for some η. If not, repeat the computation with ψ=∫_M χ(x)w∂tw for a smooth χ compactly supported in ω and equal to 1 on an interior subdomain; verify whether the resulting estimates close. If the repair works, the gap is repairable; if not, the quasi-stability proof fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Lemma 4.2(3), for ψ(t)=∫_ω w(t)∂tw(t)dx, differentiating and using (4.19) gives ψ'(t)=‖∂tw‖²_{L²(ω)}−‖∇w‖²_{L²(ω)} + ∫_{∂ω∩M} w ∂νw dS + ∫_ω (p0|w|²+p1 w∂tw)dx, up to the boundary part on ∂M where w=0. The paper omits the interior-boundary integral. For w∈H¹_0(M), Dirichlet data on ∂M do not make w vanish on ∂ω∩M, and the normal derivative of a generic H¹ function need not lie in L²(∂ω); the full trace pairing is only controlled by the H¹(ω)-norm. The displayed estimate (4.23) and the subsequent Z-estimate in Lemma 4.4 rely on this omitted term being absent. If the boundary term can only be bounded by C‖w‖²_{H¹(ω)}, it has the same differential order as the negative term −‖∇w‖²_{L²(ω)} and cannot be absorbed using only the observability inequality's lower bound without additional geometric control on Ω. This leaves the quasi-stability inequality (4.18), and with it the finite fractal dimension and regularity in Theorem 4.1, on an unproved footing. A standard repair would be to replace ψ by ∫_M χ(x)w∂tw with a smooth cutoff χ compactly supported in ω, at the cost of lower-order terms; this repair is not present in the manuscript.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, on a compact Riemannian manifold with boundary, a class of admissible ε-controllable damping regions ω whose combined interior and boundary measure is arbitrarily small (Theorem 2.1), claims that these regions satisfy the geometric control condition via an obstacle condition (Theorem 2.2), and provides an overlapping sub-domain decomposition (Theorem 2.3). Using a Carleman-based boundary observability result of Triggiani–Yao and a coarea relation, the authors derive an observability inequality with potential energy (Theorem 3.2). For a three-dimensional semilinear wave equation with locally distributed damping and critical nonlinear source, they combine this inequality with quasi-stability theory to claim existence of a finite-dimensional global attractor with H²×H¹ regularity (Theorem 4.1).","tokens_in":21292,"tokens_out":13798,"duration_ms":131984,"significance":"If the proof gaps are filled, the main result would be a genuine extension of the Chueshov–Lasiecka–Toundykov / Feireisl–Zuazua theory: it would allow damping regions of arbitrarily small interior plus boundary measure on general compact manifolds, reduce the regularity assumption on f to C¹, and still obtain finite fractal dimension and regularity. The geometric construction in Theorem 2.1 is explicit and self-contained, and the idea of converting boundary observability into a volume observability with a coarea formula is well suited to the quasi-stability framework. The observability inequality in the form (3.11) and the unique continuation consequence are potentially reusable tools. The paper is analytic and does not rely on fitted parameters; its main claims are sharply formulated and falsifiable.","major_comments":[{"comment":"Differentiating ψ(t)=∫_ω w(t)∂_t w(t) dx and integrating by parts in (4.19) produces the boundary integral ∫_{∂ω∩int M} w ∂_ν w dS, which is missing from the displayed estimate. Since the Dirichlet condition only gives w=0 on ∂M, the interior part ∂ω∩int M is not controlled, and for weak solutions w∈H¹(ω) the normal derivative need not have a trace in L²(∂ω). This term has the same differential order as the negative term -‖∇w‖²_{L²(ω)}, so it cannot be absorbed by the observability inequality without additional geometric assumptions on ω and additional regularity of the solutions. The subsequent Z-estimate in Lemma 4.4 and the quasi-stability inequality (4.18) depend on this step. A standard cutoff-function repair is not included.","section":"§4.4, Lemma 4.2(3)"},{"comment":"The definition of Z after Lemma 4.2 does not follow from (4.21) and Lemma 4.2. From those estimates one obtains a term -(η/2)‖∇w‖²_{L²(M)} - ‖∇w‖²_{L²(ω)} from the φ and ψ derivatives, not -3/2‖∇w‖²_{L²(ω)}; the lower-order potential terms are also combined inconsistently. Since Lemma 4.4 relies on the displayed form of Z to obtain the negative term that makes γ_T<1 in (4.24), the quasi-stability proof is incomplete unless the correct expression for Z is derived and verified.","section":"§4.4, proof of Theorem 4.4 (definition of Z)"},{"comment":"Theorem 3.1 is stated as a revisited version of [35, Theorem 10.1.1] with p0∈L²(0,T;L²(M)) and a finite family of overlapping sub-domains, but the only justification is the sentence that a careful revision of the proof shows this. This extension is load-bearing: the observability inequality (3.11) and the unique continuation property in Theorem 3.2 are used in Theorem 4.3 and Lemma 4.4 precisely with potentials of the form f′(u_k) and f′(αu_1+(1-α)u_2), which satisfy (3.2)-(3.3) rather than the L∞ condition in [35]. A proof or a precise reference establishing the stated version is required.","section":"§3.1, Remark 3.1"},{"comment":"The proof ends by saying that 'it is enough to take unit partition over each ~d_j' to extend the functions and obtain (5). A partition of unity does not preserve the lower bound ∇²d_j(X,X) ≥ |X|²_g or the positivity of inf_{Ω_j}|∇d_j|, and the estimates are only proven on the smaller sets W_j or ~W_j. Since Theorem 2.3 is the mechanism that supplies the overlapping sub-domains required by Theorem 3.1, this is a central gap in the observability argument.","section":"§2.3, proof of Theorem 2.3"},{"comment":"The proof asserts, without proof, that for every admissible ω there exist smooth compact submanifolds Ω_j⊂V_j with V_j\\Ω_j⊂ω and ∂Ω_j∩int M⊂ω. This existence is not part of Theorem 2.1 or of the definition of [ω_ε], and the subsequent reduction to the obstacle and escape potential conditions depends on it. As written, the claim that every admissible region satisfies (GCC) is not established.","section":"§2.2, proof of Theorem 2.2"},{"comment":"In Step 1 of the proof of Lemma 3.1, it is assumed that each connected component of ω∩∂M lies entirely in a single chart and is the level set x_N=0 of a prism P_h(Γ^j_1) contained in ω. This is not justified for arbitrary admissible ε-controllable sets; connected components of an open set on ∂M need not be contained in one coordinate chart. The proof also asserts meas_{∂M} Γ̂1 < ε0, whereas ε-controllability only gives meas_{∂M}(ω∩∂M)<ε. Since the coarea inequality (3.8) is the bridge to the volume observability (3.11), this needs a careful justification or a finite-cover argument.","section":"§3.2, proof of Lemma 3.1"}],"minor_comments":[{"comment":"The inequalities are written as ∇²d_j(X,X) ≥ |X|_g, but the scaling of the Hessian requires the square |X|²_g, as already stated in Theorem 2.1(d2).","section":"Theorem 2.3(4) and Theorem 3.1(1)"},{"comment":"The condition µ>max{2/(a0m1), 2√λ1} does not guarantee β1=µ−2/√λ1>0 when λ1<1; the proof needs the condition µ>2/√λ1 instead of, or in addition to, 2√λ1.","section":"§4.4, proof of Theorem 4.4"},{"comment":"The escape potential condition in Definition 2.5 requires |∇d|≤T/2, but the proof does not explicitly verify this bound before fixing T_j; it can be achieved by taking T_j large since d∈C∞(M), but this step should be stated.","section":"§2.2, proof of Theorem 2.2"},{"comment":"The phrase 'C1-forces with critical Sobolev growth' should be 'C¹ forcing terms with critical Sobolev growth' for consistency with the assumptions in Section 4.1.","section":"Abstract and Introduction"}],"recommendation":"major_revision","confidential_remarks":"This is a serious contribution with a clear and attractive program, but the proof gaps are concentrated in the central technical steps: the observability inequality rests on an asserted extension of Triggiani–Yao and on unproved geometric constructions, while the quasi-stability proof contains an omitted boundary term and an inconsistent auxiliary functional. These issues appear fixable within the scope of the manuscript, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nHere is my read on Cavalcanti–Ma–Marín-Rubio–Seminario-Huertas, arXiv:1908.04814.\n\nThe paper's genuine contribution is a purely geometric construction of damping regions with arbitrarily small interior plus boundary measure on compact Riemannian manifolds with boundary, plus a new observability inequality in terms of potential energy. The bridge to the geometric control condition via an obstacle condition is elegant and, if the details check out, it implies that many known stabilization results extend to these sharp measure-controlled regions. The attractor theorem for critical semilinear waves with only C^1 nonlinearities would be a real improvement over earlier C^2 results.\n\nThe geometry and the observability program are credible. The coarea argument relating boundary integrals to integrals over omega is neat. The proof of the gradient structure via unique continuation is sound.\n\nSoft spots are concentrated in the quasi-stability section. Lemma 4.2(3) drops the boundary term \\int_{\\partial\\omega\\cap M} w \\partial_\\nu w when differentiating psi(t)=\\int_\\omega w\\partial_t w. That term does not vanish for H^1 functions with Dirichlet data on \\partial M; it only vanishes on the \\partial M portion of \\partial\\omega. Without it, the claimed estimate (4.23) is not established, and the differential inequality for Phi—and hence the quasi-stability estimate (4.18)—does not follow. This is load-bearing for the finite fractal dimension and regularity in Theorem 4.1. The standard fix is to replace psi by \\int_M chi w\\partial_t w with a smooth cutoff chi compactly supported in omega, which produces only lower-order terms. I expect it is repairable, but it is not in the manuscript.\n\nTwo more issues, smaller but worth noting. Remark 3.1 asserts without proof that the Triggiani–Yao Carleman observability extends to p0 in L^2(0,T;L^2(M)) and to a finite family of overlapping subdomains. That extension is a key input; it is probably routine, but as written it is an assertion. And in the proof of Theorem 2.2, the existence of submanifolds Omega_j with V_j\\Omega_j subset omega is asserted \"by construction\" without a real argument; the admissible class [omega_epsilon] is not pinned down enough to make the GCC claim fully rigorous.\n\nThe paper is not circular; the self-citations provide tools, not the conclusion. It is clearly written and the overall strategy is sound.\n\nI would send it to a serious referee. The geometric result is worth publishing even if the attractor theorem needs another round, and the referee should push for the boundary-term fix before acceptance.\n\nBest,\n\n[Your name]","headline":"Genuinely new geometric construction and a promising potential-energy observability inequality, but the quasi-stability proof drops a boundary term and the attractor theorem is not yet fully established.","tokens_in":21855,"tokens_out":4558,"would_cite":false,"duration_ms":45076,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B40","35L20","35B41","93B07","35R01"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs damping regions whose interior and boundary measures can be made arbitrarily small, yet still control the wave equation and yield finite-dimensional global attractors.","keywords":["observability","unique continuation","localized damping","Riemannian manifold","global attractor","geometric control condition","quasi-stability","critical Sobolev growth"],"falsifier":"Search for a potential $p_0\\in L^2(0,T;L^2(M))$ and a solution of $\\partial_t^2 w-\\Delta w=p_0 w+p_1\\partial_t w$ on a compact manifold with boundary for which the boundary observability inequality (3.5) fails while the hypotheses of Theorem 3.1 hold; equivalently, exhibit a nonzero solution that vanishes to infinite order on $\\Gamma_1\\times(0,T)$. Such a counterexample would invalidate the extension claimed in Remark 3.1 and, with it, the attractor theorem.","tokens_in":20742,"feed_emoji":"🌊","tokens_out":11197,"duration_ms":94709,"temperature":0.7,"pith_summary":"The paper aims to show that the damping region for a semilinear wave equation on a compact Riemannian manifold with boundary can be made extremely small—both inside the manifold and along its boundary—without destroying the long-time dynamics. It constructs, for any prescribed $\\varepsilon>0$, an open set $\\omega$ whose summed interior and boundary measure is less than $\\varepsilon$, proves these $\\varepsilon$-controllable regions satisfy the geometric control condition, and establishes a new observability inequality in terms of potential energy on $\\omega$. Using that inequality together with unique continuation, it then proves that, on a three-dimensional manifold, the semilinear wave equation with locally distributed damping on such a region and a critical nonlinearity has a global attractor with finite fractal dimension and $H^2(M)\\times H^1(M)$ regularity. The interest is that previous results in the Euclidean setting required stronger regularity of the nonlinearity, while here merely $C^1$ is enough, and the measure-sharpness is purely geometric.","feed_headline":"Arbitrarily small dampers yield finite-dimensional wave attractors","feed_subtitle":"New construction makes damping regions with tiny interior and boundary measure work for critical wave equations on curved spaces.","key_machinery":"Two pieces of machinery carry the argument. The first is the geometric construction of a 'scape' potential $d$ on a large subset $V\\subset M$: on $V$ one has $\\mathrm{Hess}\\, d(X,X) \\ge |X|^2_g$, $\\inf_V |\\nabla d|>0$, and $\\langle\\nabla d,\\nu\\rangle<0$ on $\\partial M\\cap V$, while $M\\setminus V$ has measure $\\varepsilon_0<\\varepsilon$. Since $d$ has no critical points on $V$, the boundary region $\\Gamma_1=\\{x\\in\\partial M:\\langle\\nabla d,\\nu\\rangle>0\\}$ provides the observation set required by a Carleman-based boundary observability theorem, quoted and extended in Theorem 3.1. The second ingredient is a coarea-type relation (Lemma 3.1) that converts the boundary integral over $\\Gamma_1$ into a volume integral over $\\omega$ with a constant $C(\\varepsilon)$: $\\int_{\\Gamma_1} f\\,d\\sigma \\le C\\int_\\omega f\\,dg$. Feeding $f=|\\nabla w|^2$ through this relation converts boundary observability into the volume potential-energy inequality (3.11), which is exactly what the quasi-stability theory needs. The attractor result then follows from the gradient structure (unique continuation) and quasi-stability estimates that use Strichartz estimates for the critical cubic nonlinearity.","core_discovery":"The central discovery is that a control/damping region whose total measure—interior plus boundary—is arbitrarily small can nevertheless be fully effective for wave dynamics. Concretely, given $\\varepsilon>0$ the authors construct an open set $V$ with smooth boundary, a 'scape' potential $d$ with $\\mathrm{Hess}\\, d \\ge g$ on $V$, and an open $\\omega$ containing $M\\setminus V$ such that $\\mathrm{meas}_M(\\omega)+\\mathrm{meas}_{\\partial M}(\\omega\\cap\\partial M)<\\varepsilon$. For such admissible regions they prove the observability inequality $$\\int_0^T\\int_\\omega |\\nabla w|^2\\,dx\\,dt \\ge k_T\\big(\\|(w(0),\\partial_t w(0))\\|$_H^{2}$+\\|(w(T),\\partial_t w(T))\\|$_H^{2}$\\big),$$ for solutions of the linear wave equation with potentials, together with a unique continuation property. In three dimensions, under assumptions (4.2)–(4.4) and $a\\ge a_0>0$ a.e. on an admissible region $\\omega$, the semilinear wave equation $\\partial_t^2 u-\\Delta u+a(x)g(\\partial_t u)+f(u)=0$ has a global attractor $A$ with finite fractal dimension and regularity $H^2(M)\\times H^1(M)$. The proof combines the observability inequality with the quasi-stability theory, using Strichartz estimates to handle the critical exponent.","pith_inferences":["Editorial inference: the geometric construction (Theorem 2.1 and the GCC bridge) is dimension-free, so admissible $\\varepsilon$-controllable regions exist on manifolds of any dimension; only the attractor step uses three-dimensional Strichartz estimates, so the same regions could feed controllability results in higher dimensions.","Editorial inference: since the coarea relation converts boundary integrals into volume integrals over $\\omega$ with a constant independent of $f$, the same potential-energy observability should hold for wave equations with more general lower-order terms than the Lipschitz damping treated here, potentially broadening the class of admissible nonlinearities.","Editorial inference: because the boundary measure of admissible regions can be made arbitrarily small, a natural next problem is boundary stabilization—a damper supported on a boundary set of arbitrarily small measure—provided a boundary version of the observability inequality can be proved."],"forward_implications":["Damping can be confined to regions of arbitrarily small total measure (interior plus boundary) without sacrificing exponential stabilization or controllability, because every admissible region satisfies the geometric control condition; existing GCC-based results therefore transfer to this measure-sharp setting.","The semilinear wave equation with critical cubic growth and merely $C^1$ nonlinearities has a finite-dimensional, $H^2(M)\\times H^1(M)$-regular global attractor under localized damping on an $\\varepsilon$-controllable region, so its long-time dynamics are captured by finitely many modes.","The new observability inequality is expressed in terms of potential energy rather than kinetic energy and holds for potentials $p_0\\in L^2(0,T;L^2(M))$, exactly the regularity needed for the quasi-stability proof.","The accompanying unique continuation property—vanishing on $\\omega\\times(0,T)$ forces vanishing everywhere—is a stand-alone tool for inverse problems and exact controllability of waves with lower-order terms."],"supporting_citations":[{"why":"Establishes the sharp geometric control condition for wave observability that the constructed regions are shown to satisfy.","marker":"[3]"},{"why":"Supplies the base geometric construction of escape functions on compact manifolds with boundary, extended in Theorem 2.1.","marker":"[7]"},{"why":"Provides the Carleman boundary observability theorem quoted as Theorem 3.1 that the paper revisits and extends.","marker":"[35]"},{"why":"Introduces the quasi-stability framework used to derive finite fractal dimension of attractors.","marker":"[10]"},{"why":"Provides the attractor existence and fractal dimension theorems used to conclude the main result.","marker":"[12]"},{"why":"Gives the Euclidean $C^2$-prior result that the present paper extends to manifolds with $C^1$ nonlinearities.","marker":"[11]"},{"why":"Supplies the escape function and geodesic condition bridge used to prove that admissible regions satisfy GCC.","marker":"[25]"},{"why":"Supplies the Strichartz estimates used to control the critical cubic nonlinearity in the quasi-stability proof.","marker":"[32]"}],"fun_headline_variants":["Tiny damping regions control Riemann wave dynamics","Arbitrarily small dampers give finite-dimensional attractors","Measure-controlled damping: sharp regions, full wave control","Potential-energy observability from tiny damping sets","Wave attractors on 3D manifolds with tiny damping"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the Carleman boundary observability theorem, claimed without proof to extend to potentials $p_0\\in L^2(0,T;L^2(M))$ and to finite overlapping subdomains, and the asserted existence of submanifolds $\\Omega_j\\subset V_j$ with $V_j\\setminus\\Omega_j\\subset\\omega$ in the proof of Theorem 2.2, hold exactly as stated.","fun_headline_variants_meta":{"raw":{"variants":["Tiny damping regions control Riemann wave dynamics","Arbitrarily small dampers give finite-dimensional attractors","Measure-controlled damping: sharp regions, full wave control","Potential-energy observability from tiny damping sets","Wave attractors on 3D manifolds with tiny damping"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000292,"raw_usage":{"total_tokens":1735,"prompt_tokens":1006,"completion_tokens":729,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":622,"completion_tokens_details":{"reasoning_tokens":653}},"tokens_in":622,"tokens_out":729,"duration_ms":7442,"temperature":1.0,"reasoning_tokens":653,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:33:30.031073+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search for a potential $p_0\\in L^2(0,T;L^2(M))$ and a solution of $\\partial_t^2 w-\\Delta w=p_0 w+p_1\\partial_t w$ on a compact manifold with boundary for which the boundary observability inequality (3.5) fails while the hypotheses of Theorem 3.1 hold; equivalently, exhibit a nonzero solution that vanishes to infinite order on $\\Gamma_1\\times(0,T)$. Such a counterexample would invalidate the extension claimed in Remark 3.1 and, with it, the attractor theorem.","supporting_citations":[{"cited_title":"Bardos, G","cited_arxiv_id":null,"evidence_quote":"Establishes the sharp geometric control condition for wave observability that the constructed regions are shown to satisfy."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the base geometric construction of escape functions on compact manifolds with boundary, extended in Theorem 2.1."},{"cited_title":"Triggiani and P","cited_arxiv_id":null,"evidence_quote":"Provides the Carleman boundary observability theorem quoted as Theorem 3.1 that the paper revisits and extends."},{"cited_title":"Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Universitext, Springer, Cham, 2015","cited_arxiv_id":null,"evidence_quote":"Introduces the quasi-stability framework used to derive finite fractal dimension of attractors."},{"cited_title":"Chueshov and I","cited_arxiv_id":null,"evidence_quote":"Provides the attractor existence and fractal dimension theorems used to conclude the main result."},{"cited_title":"Chueshov, I","cited_arxiv_id":null,"evidence_quote":"Gives the Euclidean $C^2$-prior result that the present paper extends to manifolds with $C^1$ nonlinearities."},{"cited_title":"Miller, Escape function conditions for the observation, control, and stabi- lization of the wave equation , SIAM J","cited_arxiv_id":null,"evidence_quote":"Supplies the escape function and geodesic condition bridge used to prove that admissible regions satisfy GCC."},{"cited_title":"Strichartz, Restriction of Fourier transform to quadratic surfaces and de- cay of solutions to the wave equation , Duke Math","cited_arxiv_id":null,"evidence_quote":"Supplies the Strichartz estimates used to control the critical cubic nonlinearity in the quasi-stability proof."}],"review_version":1}