{"id":"98c65e57-0d23-4b7f-9817-79b00ef0e1c5","arxiv_id":"1908.02987","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the 2D inhomogeneous NLS with 0<b<1 and α>2-b, radial H^1 solutions scatter in both focusing (below ground state) and defocusing cases.","lead":"This paper proves energy scattering for a two-dimensional inhomogeneous nonlinear Schrödinger equation with radial initial data, extending the known range of the inhomogeneity parameter b from below 2/3 to the full range b<1. The result covers both focusing solutions below the ground state threshold and all defocusing solutions, using a recent proof strategy by Arora, Dodson and Murphy.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.2's bootstrap uses αθ/(α+2+b)>1/2, but the θ from Lemma 3.1 gives the opposite; the ε^{1/2} bound for (3.8) is not established as written.","rationale":"The reader identified radial symmetry as the weakest assumption, but radial symmetry is a hypothesis of the theorem and is used consistently. My review focuses on a different, more concrete defect: the proof of the key spacetime bound Proposition 3.2 contains a false exponent comparison. Lemma 3.1 produces a θ whose associated exponent αθ/(α+2+b) is always below 1/2, yet Proposition 3.2 relies on the opposite to justify the ε^{1/2} smallness in the bootstrap. This is not a stylistic issue: (3.8) is the step that makes the linear evolution small on the interval [t1,e], which is essential for the continuity argument. The gap appears repairable because the bootstrap only needs the smallness δ→0 as ε→0, which holds for any p>0, and one can compensate by taking T larger. However, as written, the proof of Proposition 3.2 is incomplete. Since the central claim is likely true but the written proof requires correction, I recommend a conditional acceptance rather than unconditional acceptance. I did not find other issues of comparable weight: the Morawetz estimates, the radial Sobolev use, the variational lemmas, and the final scattering argument are standard and appear sound under the stated hypotheses.","tokens_in":22532,"tokens_out":41353,"duration_ms":360246,"concrete_test":"Recompute the exponent p=αθ/(α+2+b) from the θ formula in Lemma 3.1 at an admissible parameter point, e.g., b=0.5, α=3, taking η>0 small and letting ε→0. The limiting value is (2-b-η)/4<0.5, which directly contradicts the assertion in Proposition 3.2 that αθ/(α+2+b)>1/2. If this computation is confirmed, check whether replacing ε^{1/2} in (3.8) by ε^p with p=αθ/(α+2+b) still allows the continuity argument in (3.9) to close, with T chosen so large that (εT^{1−β1−β2})^{−δ}≤ε^p.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Proposition 3.2, the contribution from [t0,t1] to the tail Duhamel integral is bounded via (3.2) by C ε^{αθ/(α+2+b)}, using the short-interval estimate (3.11). The proof then asserts that, by taking ∞−=1/ε in Lemma 3.1 with ε>0 sufficiently small, αθ/(α+2+b)>1/2, so this contribution is O(ε^{1/2}). This assertion is false for the θ constructed in Lemma 3.1. Indeed, the unit-ball estimate in Lemma 3.1 requires 2/ν=b+η with η>0, and the displayed θ satisfies θ(ε) = (2-b-η-2αε)/(4α/(α+2+b)-2αε). Its limit as ε→0 is θ0=(α+2+b)(2-b-η)/(4α), hence αθ0/(α+2+b)=(2-b-η)/4, which is strictly less than 1/2 for all 0<b<1 and η>0. Thus the exponent p:=αθ/(α+2+b) is bounded away from 1/2 (actually p≤(2-b)/4<1/2), so the ε^p term cannot be absorbed into C ε^{1/2} as ε→0. Consequently, the proof of (3.8) as written does not close. The theorem may still be true—one can rerun the bootstrap with δ=ε^p and choose T larger, since p>0—but the written argument contains a concrete, load-bearing false estimate at the heart of Proposition 3.2.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves energy scattering for radial H^1 solutions of the two-dimensional inhomogeneous nonlinear Schrödinger equation i∂_t u + Δu = ±|x|^{-b}|u|^α u, in the L^2-supercritical range 0<b<1, α>2-b. In the focusing case, Theorem 1.3 assumes the standard ground-state threshold (1.6)-(1.7) and extends earlier results from b<2/3 to b<1; in the defocusing case, Theorem 1.6 establishes scattering for all radial H^1 data. The proof combines Morawetz/virial estimates that yield polynomial-in-time space-time bounds (Proposition 2.6 and Corollary 2.7) with an Arora-Dodson-Murphy type bootstrap (Proposition 3.2) producing a global L^{α+2+b}(R×R^2) bound, from which scattering follows via Strichartz estimates. An appendix gives an alternative interaction-Morawetz proof for the defocusing problem in dimensions N≥3.","tokens_in":22844,"tokens_out":16853,"duration_ms":166454,"significance":"If the proofs were correct as written, the main results would be a solid extension of the scattering theory for the inhomogeneous NLS and a useful alternative to concentration-compactness arguments. The paper is carefully organized, gives explicit Morawetz estimates, and correctly leverages prior variational lemmas (Lemmas 2.3 and 2.4) and Strichartz technology. However, the central bootstrap step in Proposition 3.2 contains a quantitative assertion that is false: the claimed ε^{1/2} smallness in (3.8) is not established. Because this step drives both the focusing and the defocusing theorems, the manuscript needs a substantive revision. The error appears repairable by re-running the bootstrap with a different smallness exponent, so the underlying approach remains credible.","major_comments":[{"comment":"The proof asserts that by taking ∞−=1/ε in Lemma 3.1 one obtains αθ/(α+2+b)>1/2. This is incompatible with the θ constructed in Lemma 3.1. For the unit-ball term, θ(ε)=(2-b-η-2αε)/(4α/(α+2+b)-2αε), hence αθ(ε)/(α+2+b)=(2-b-η-2αε)/(4-2ε(α+2+b)) → (2-b-η)/4 as ε→0, which is strictly below 1/2 for every η>0 and 0<b<1. The complementary exterior term gives the limit (2-b+η)/4, which is also below 1/2 because the condition 2/ν=b-η forces η<b. Therefore the contribution of [t0,t1] is bounded by C ε^{αθ/(α+2+b)} with exponent smaller than 1/2, and the displayed bound in (3.8) does not follow. This is a load-bearing step for Proposition 3.2 and hence for Corollary 3.3 and Theorems 1.3 and 1.6. The argument can likely be repaired by replacing ε^{1/2} with ε^p for p=(2-b-η)/4 (or an analogous power) and choosing ε small enough for the subsequent continuity argument, but the manuscript as written contains a false estimate at the core of the proof.","section":"Proposition 3.2, step leading to (3.8)"}],"minor_comments":[{"comment":"The displayed formula for θ in Lemma 3.1 is missing parentheses and is easy to misread; it should be written as θ(ε)=(2-b-η-2αε)/(4α/(α+2+b)-2αε) with explicit brackets.","section":"Lemma 3.1"},{"comment":"The notation ∞−=1/ε in Proposition 3.2 reuses the same symbol ε as the small parameter chosen at the start of the proof; these are logically independent and should be denoted by different symbols, especially since the erroneous exponent claim in (3.8) depends on the confusion.","section":"Proposition 3.2"},{"comment":"Remark 1.4 says the result extends to the whole range of b where local well-posedness is available, but in two dimensions local well-posedness is available for all α>0, while Theorem 1.3 imposes α>2-b; the remark should be phrased as the whole range of b for the L^2-supercritical regime.","section":"Remark 1.4"},{"comment":"The proofs end with the literal token \"/Box\" instead of the usual end-of-proof symbol; this is a formatting artifact that should be corrected.","section":"Proofs of Theorem 1.3 and Theorem 1.6"}],"recommendation":"major_revision","confidential_remarks":"The main issue is localized to Proposition 3.2 and appears repairable, so I recommend major revision rather than rejection. The author also cites simultaneous independent work by Xu-Zhao for the focusing result, so the novelty is mostly in the method and in the defocusing 2D result; this is not a reason for rejection, but the referee report should focus on the quantitative gap in the bootstrap."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. The result is a real extension: focusing scattering for the 2D inhomogeneous NLS in the full local well-posedness range 0<b<1 (previous work stopped at b<2/3), plus a first 2D defocusing radial scattering theorem. The method, adapting the Arora-Dodson-Murphy bootstrap with Morawetz estimates, is well chosen and mostly cleanly executed. But the proof of Proposition 3.2 has a load-bearing false estimate. As written, the bootstrap does not close.\n\nWhat is genuinely new: Farah-Guzmán did b<2/3, and the author pushes to b<1 using the ADM approach. The defocusing 2D radial case is also new. The Morawetz estimates in Section 2 are handled carefully, with the radial Sobolev embedding doing real work on the annulus. The paper is honest about the simultaneous independent proof by Xu-Zhao. Citation patterns are fine; self-citations point to relevant prior results.\n\nThe soft spot is not a detail. In Lemma 3.1, the exponent θ is constrained by the unit-ball estimate: writing 2/ν=b+η with η>0, the displayed θ satisfies αθ/(α+2+b) → (2−b−η)/4 as the auxiliary ε→0. For 0<b<1 this limit is strictly below 1/2. So the assertion in Proposition 3.2 that “αθ/(α+2+b) > 1/2” for small ε is false. Consequently the ε^{αθ/(α+2+b)} term in the final estimate cannot be absorbed into the desired ε^{1/2} bound (3.8). The theorem may still be true—one can rerun the bootstrap with ε^p for p=αθ/(α+2+b)>0 and choose T larger, since αθ>1—but that is a substantive repair, not a typo. A referee should insist on it.\n\nThe reader’s report missed this; soundness is lower than the given 7. The reliance on external lemmas (Farah-Guzmán, Campos) is standard and not a concern.\n\nThis paper is for dispersive PDE people working on scattering for inhomogeneous NLS. The result is worth having, but the proof needs a real revision before publication. Send it to a serious referee, with expectation of a major revision.","headline":"Genuine extension of the 2D INLS scattering result, but Proposition 3.2 contains a load-bearing false exponent estimate that the proof, as written, cannot absorb.","tokens_in":23428,"tokens_out":6945,"would_cite":true,"duration_ms":68281,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q55","35Q44"],"pacs":[],"model":"deepseek-v4-flash","headline":"Radial 2D NLS solutions scatter in both time directions","keywords":["inhomogeneous nonlinear Schrödinger equation","energy scattering","radial symmetry","Morawetz estimates","ground state threshold","Strichartz estimates","two dimensions"],"falsifier":"Numerically simulate the focusing 2D equation with radial data at, say, $b=0.8$ and $\\alpha=1.3$ satisfying (1.6) and (1.7), and monitor $\\|u\\|_{L^{\\alpha+2+b}([-T,T]\\times\\mathbb{R}^2)}$; if this quantity fails to stay bounded while the solution remains smooth, the global bound (3.4) and hence the scattering conclusion would be false.","tokens_in":22297,"feed_emoji":"⚛️","tokens_out":8988,"duration_ms":95572,"temperature":0.7,"pith_summary":"The paper establishes energy scattering for a two-dimensional inhomogeneous nonlinear Schrödinger equation with nonlinearity $\\pm |x|^{-b}|u|^\\alpha u$, where $0<b<1$ and $\\alpha>2-b$, for radially symmetric $H^1$ initial data. In the focusing case, any radial datum satisfying the ground-state threshold conditions (1.6) and (1.7) gives a global solution that converges to a free wave in $H^1$ as $t\\to\\pm\\infty$. In the defocusing case, the same scattering conclusion holds for every radial $H^1$ datum. The proof works by deriving Morawetz estimates that yield polynomial-in-time space-time bounds, then bootstrapping these into a global $L^{\\alpha+2+b}(\\mathbb{R}\\times\\mathbb{R}^2)$ bound, which forces the solution to approach linear evolution. This settles the focusing 2D case across the full range of $b$ for which local well-posedness is available, extending previous results that were restricted to $b<2/3$.","feed_headline":"Radial 2D NLS solutions scatter in both time directions","feed_subtitle":"A weighted space-time bound forces convergence to free waves for focusing and defocusing data.","key_machinery":"The load-bearing machinery is a radial Morawetz estimate, Proposition 2.6, derived from a virial identity with a cutoff radial weight $\\phi_R$. Radial Sobolev embedding controls the contribution from the annulus $\\{R\\le |x|\\le 2R\\}$, and a localized variational lemma keeps the difference between kinetic energy and weighted nonlinear energy positive below the ground-state threshold. The result is the polynomial bounds $\\int_0^T\\int |x|^{-b}|u|^{\\alpha+2}\\,dx\\,dt\\lesssim T^{\\beta_1}$ and $\\int_I\\|u(t)\\|_{L^{\\alpha+2+b}}^{\\alpha+2+b}\\,dt\\lesssim |I|^{\\beta_2}$, with $\\beta_1+\\beta_2<1$. These feed a bootstrap argument that upgrades the local-in-time bounds to the global space-time bound from which scattering follows.","core_discovery":"The central discovery is that radial symmetry, together with the ground-state subthreshold conditions in the focusing case, forces a finite global space-time norm: $\\|u\\|_{L^{\\alpha+2+b}(\\mathbb{R}\\times\\mathbb{R}^2)}\\le C(u_0,Q)<\\infty$. Once this global bound is available, a Strichartz argument shows that the full nonlinear solution has finite Strichartz norm, and consequently the limits defining the scattering states $u_0^\\pm$ exist in $H^1$ and $u(t)-e^{it\\Delta}u_0^\\pm\\to0$. Theorems 1.3 and 1.6 state exactly this for the focusing and defocusing problems, respectively, in two dimensions with $0<b<1$ and $\\alpha>2-b$.","pith_inferences":["The Morawetz machinery in Proposition 2.6 is written for all $N\\ge2$, so the method is likely to extend to higher-dimensional radial focusing INLS with the same type of subthreshold conditions, with exponents adjusted for the dimension.","The polynomial space-time bounds may carry quantitative information, potentially yielding explicit decay rates for the nonlinear term along the solution and hence rates of convergence to the scattering states, not merely qualitative convergence.","Radial symmetry is used only to control the annulus contribution through the radial Sobolev embedding; replacing that control by an angular-averaged or nonradial argument, if possible, would open the way to nonradial 2D scattering below the same threshold.","The threshold conditions (1.6) and (1.7) enter only through uniform coercivity and the localized variational lemma, so one can test whether the scattering conclusion remains true for perturbations of the ground state or for nearby thresholds."],"forward_implications":["In the focusing case, scattering below the ground state now holds for every $b\\in(0,1)$ in two dimensions, the full range where the local theory used here is available.","In the defocusing case, radial $H^1$ data in two dimensions scatter, extending energy scattering to a setting not covered by the previous nonradial dimensions-$N\\ge3$ result.","The global bound $\\|u\\|_{L^{\\alpha+2+b}(\\mathbb{R}\\times\\mathbb{R}^2)}<\\infty$ implies the global Strichartz bound $\\|\\langle\\nabla\\rangle u\\|_{S(L^2,\\mathbb{R})}<\\infty$, which is exactly what secures existence of the wave operators.","The proof avoids concentration-compactness arguments, giving a direct route from Morawetz estimates to scattering in the radial 2D focusing case.","The same mechanism covers both signs of the nonlinearity, so the scattering phenomenon is not tied to the defocusing sign."],"supporting_citations":[{"why":"Supplies the bootstrap strategy that converts Morawetz-style polynomial bounds into a global space-time bound.","marker":"[1]"},{"why":"Provides the localized variational lemma used to keep the Morawetz differential inequality coercive below the threshold.","marker":"[3]"},{"why":"Gives the Strichartz estimates used throughout the well-posedness and scattering arguments.","marker":"[4]"},{"why":"Establishes defocusing energy scattering in dimensions $N\\ge3$, the result the 2D defocusing theorem extends.","marker":"[10]"},{"why":"Provides the new proof of scattering below the ground state that the present argument adapts to the inhomogeneous setting.","marker":"[12]"},{"why":"Supplies global well-posedness below the ground-state threshold and the virial identity underlying the Morawetz estimates.","marker":"[13]"},{"why":"Contains the prior 2D focusing scattering result with restricted $b$ and the variational lemmas reused here.","marker":"[15]"},{"why":"The radial Sobolev embedding used to control the annulus contribution in the Morawetz estimate.","marker":"[25]"},{"why":"Provides the Strichartz theory and nonlinear estimates used to turn the global bound into scattering.","marker":"[26]"}],"fun_headline_variants":["Radial data guarantee energy scattering for 2D inhomogeneous NLS","Energy scattering proved for radial 2D NLS with focusing and defocusing","2D inhomogeneous NLS: radial symmetry yields scattering","Scattering in 2D INLS for radial initial data","Radial 2D NLS scattering in both signs of nonlinearity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the initial data being radially symmetric, which is what controls the annulus term in the Morawetz estimate, and, in the focusing case, on the data satisfying the ground-state threshold conditions (1.6) and (1.7).","fun_headline_variants_meta":{"raw":{"variants":["Radial data guarantee energy scattering for 2D inhomogeneous NLS","Energy scattering proved for radial 2D NLS with focusing and defocusing","2D inhomogeneous NLS: radial symmetry yields scattering","Scattering in 2D INLS for radial initial data","Radial 2D NLS scattering in both signs of nonlinearity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000366,"raw_usage":{"total_tokens":1941,"prompt_tokens":888,"completion_tokens":1053,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":504,"completion_tokens_details":{"reasoning_tokens":960}},"tokens_in":504,"tokens_out":1053,"duration_ms":10630,"temperature":1.0,"reasoning_tokens":960,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:27:39.647391+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically simulate the focusing 2D equation with radial data at, say, $b=0.8$ and $\\alpha=1.3$ satisfying (1.6) and (1.7), and monitor $\\|u\\|_{L^{\\alpha+2+b}([-T,T]\\times\\mathbb{R}^2)}$; if this quantity fails to stay bounded while the solution remains smooth, the global bound (3.4) and hence the scattering conclusion would be false.","supporting_citations":[{"cited_title":"Scattering below the ground state for the 2$d$ radial nonlinear Schr\\\"odinger equation","cited_arxiv_id":"1906.00515","evidence_quote":"Supplies the bootstrap strategy that converts Morawetz-style polynomial bounds into a global space-time bound."},{"cited_title":"Scattering of radial solutions to the Inhomogeneous Nonlinear Schr\\\"odinger Equation","cited_arxiv_id":"1905.02663","evidence_quote":"Provides the localized variational lemma used to keep the Morawetz differential inequality coercive below the threshold."},{"cited_title":"Cazenave, Semilinear Schr¨ odinger Equations, Courant Lecture Notes in Mathematics 10, American Mathema tical Society, Courant Institute of Mathematical Sciences, 2003","cited_arxiv_id":null,"evidence_quote":"Gives the Strichartz estimates used throughout the well-posedness and scattering arguments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes defocusing energy scattering in dimensions $N\\ge3$, the result the 2D defocusing theorem extends."},{"cited_title":"Dodson and J","cited_arxiv_id":null,"evidence_quote":"Provides the new proof of scattering below the ground state that the present argument adapts to the inhomogeneous setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies global well-posedness below the ground-state threshold and the virial identity underlying the Morawetz estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contains the prior 2D focusing scattering result with restricted $b$ and the variational lemmas reused here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The radial Sobolev embedding used to control the annulus contribution in the Morawetz estimate."},{"cited_title":"Tao, Nonlinear dispersive equations: local and global analysis , CBMS Regional Conference Series in Mathematics 106, AMS, 2006","cited_arxiv_id":null,"evidence_quote":"Provides the Strichartz theory and nonlinear estimates used to turn the global bound into scattering."}],"review_version":1}