A remark on the scattering theory for the 2d radial focusing INLS

Chengbin Xu; Tengfei Zhao

REVIEW 2 major objections 5 minor 1 cited by

A remark on the scattering theory for the 2d radial focusing INLS

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that radial solutions below the ground-state threshold to the 2d focusing inhomogeneous nonlinear Schrödinger equation scatter for all 0<b<1, extending the known range 0<b<2/3, and gives a proof that avoids concentration…

desk verdict Worth a serious referee: a real but incremental extension of the scattering range for 2d radial INLS, with a fixable gap in Lemma 3.1 and an unclear relation to Campos [2]. read the letter →

arxiv 1908.00743 v3 pith:2NSKW2P3 submitted 2019-08-02 math.AP

Chengbin Xu , Tengfei Zhao This is my paper · ORCID

classification math.AP MSC 35P2535Q5547J35

keywords scatteringtheoryinhomogeneousnonlinearSchrödingerequationradialsolutionsgroundstatethresholdconcentrationcompactnessMorawetzestimatesStrichartztwo-dimensionalNLS

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that for the two-dimensional focusing inhomogeneous nonlinear Schrödinger equation $i\partial_t u+\Delta u+|x|^{-b}|u|^p u=0$, every radial solution whose mass-energy product and product norm lie below the ground-state values is globally well-posed and scatters, for the full parameter range $0

What carries the argument

The load-bearing object is the Strichartz norm $S(\dot H^{s_p},I)$, defined as the supremum of $L^q_t L^r_x$ norms over all $\dot H^{s_p}$-admissible pairs, and the scattering criterion (Lemma 3.1) is phrased in that norm. The proof follows a modified scheme from [1]: it splits the Duhamel term at a large time $T_0$ into a far past piece, controlled by dispersion and the assumed bound $\int_0^T\int |x|^{-b}|u|^{p+2}\lesssim T^\alpha$, and a recent piece, controlled by radial Sobolev embedding together with smallness of the mass inside a ball. The Morawetz estimate comes from the Virial/Morawetz identity (4.18)-(4.19) applied to a radial weight $a(x)$ that grows like $|x|^2$ inside a ball and like $|x|$ outside, giving $\int_0^T\int_{\mathbb{R}^2}|x|^{-b}|u|^{p+2}\lesssim R+T/R^\alpha$; the choice $R=T^{1/(1+\alpha)}$ then yields the spacetime bound required by the scattering criterion.

What would settle it

Take any nonzero radial $u_0\in H^1$ and evaluate $\|e^{it\Delta}u_0\|_{S(\dot H^{s_p},[T_0,\infty))}$ directly: the endpoint admissible pair $(q,r)=(\infty,2/(1-s_p))$ is included in the definition, and at that pair the norm equals $\|u_0\|_{\dot H^{s_p}}$, independent of $T_0$, so the asserted smallness as $T_0\to\infty$ fails exactly as written; a corrected non-endpoint Strichartz norm would be needed for the criterion's proof to succeed.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: under $0<b<1$ and $2-b<p<\infty$, if $u_0$ is radial and satisfies $M(u_0)^{1-s_p}E(u_0)^{s_p}<M(Q)^{1-s_p}E(Q)^{s_p}$ and $\|u_0\|_{L^2}^{1-s_p}\|\nabla u_0\|_{L^2}^{s_p}<\|Q\|_{L^2}^{1-s_p}\|\nabla Q\|_{L^2}^{s_p}$, then the solution $u$ is globally well-posed and scatters, meaning there exist $u_\pm\in H^1$ with $\|u(t)-e^{it\Delta}u_\pm\|_{H^1}\to 0$ as $t\to\pm\infty$. The authors prove this by showing that such solutions satisfy a new scattering criterion: a uniform $H^1$ bound together with the weighted spacetime estimate $\int_0^T\int_{\mathbb{R}^2}|x|^{-b}|u|^{p+2}dxdt\lesssim T^\beta$, for some $0<\beta<1$, implies scattering forward in time. This provides the scattering conclusion for the full range $0<b<1$, reaching the boundary where the local well-posedness theory from [5] is available, and it removes the need for concentration compactness.

Load-bearing premise

The proof assumes that for sufficiently large $T_0$, the linear evolution $e^{it\Delta}u_0$ has small norm in the Strichartz space used by the scattering criterion; as written that space includes an endpoint admissible pair at which the norm is conserved rather than decaying, so the statement needs a non-endpoint version that is not supplied.

Editorial extensions

If this is right

Scattering below the ground state for the 2d radial focusing INLS holds for every $b\in(0,1)$ and $p\in(2-b,\infty)$, not just for the previously covered range $b<2/3$.
The Virial/Morawetz estimate $\int_0^T\int |x|^{-b}|u|^{p+2}\lesssim T^\beta$ supplies the long-time smallness of the nonlinear term directly, giving an alternate route to conclusions previously reached by concentration compactness.
The scattering criterion in Lemma 3.1 is a reusable tool: any radial solution with a uniform $H^1$ bound and the stated power-law decay of the weighted nonlinear integral will scatter.
The theorem confirms that the below-ground-state threshold expressed by $M(u)^{1-s_p}E(u)^{s_p}<M(Q)^{1-s_p}E(Q)^{s_p}$ and the product-norm condition is sufficient for scattering for the full allowed range of $b$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The endpoint issue in Lemma 3.1 suggests the intended norm may be a non-endpoint Strichartz norm; if so, the proof could likely be repaired by a standard limiting argument, but the written smallness assertion is not justified as it stands.
The same combination of a scattering criterion, radial Sobolev embedding, and a convex-weight Morawetz estimate may extend to other dimensions or to other inhomogeneous nonlinearities with decaying weights, wherever the relevant coercivity and radial decay estimates hold.
A numerical simulation of radial solutions near the threshold with $b$ close to $1$ could test the predicted power-law decay of $\int_0^T\int |x|^{-b}|u|^{p+2}$ independently of the proof, providing evidence on whether the Morawetz estimate is the right mechanism.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a worthy incremental paper with a fixable gap in Lemma 3.1. The main theorem, scattering for radial 2d focusing INLS below ground state in the full range 0

The cleanest part is the scattering criterion. It is the right kind of tool for this area, and the proof through the Duhamel split F1/F2 is mostly standard. The paper is also reasonably self-contained: the nonlinear estimates in Lemma 2.3 are stated with proof sketches, and the global well-posedness input is quoted properly.

The soft spots, in order of importance.

First, the report I saw claims the linear tail ||e^{itΔ}u0||_{S(Ḣ^{sp},[T0,∞))} cannot be small because the S norm includes a conserved H^{sp} piece. I think that objection is off. The endpoint pair is (q=∞, r=2/(1−sp)), which is L∞_t L^r_x, not L∞_t H^{sp}_x. For fixed u0∈H^{sp}, that tail tends to zero by dispersive decay after a Schwartz approximation. So this is not a genuine gap.

The genuine gap is in the F2 estimate. After fixing T0, the proof chooses T>T0 satisfying (3.15). Then it uses the derivative bound |d/dt∫χ_R|u|^2|≲1/R to conclude smallness of ||χ_R u||_{L∞_t L^2_x(I2)} on I2=[T0−ε^{-θ},T0]. But the smallness is at T, which is outside I2 and whose distance from I2 is uncontrolled. The derivative bound only propagates from an endpoint of the interval. This is fixable by choosing T0 itself from the liminf sequence in (3.10) and setting T=T0 (or an equivalent rearrangement), but as written it is a real logical gap. It should be fixed before publication.

Second, the paper cites Campos [2] without saying how Theorem 1.1 relates to it. If Campos already proved the same range, the novelty is only the method; if not, the authors should say so explicitly. This needs clarification.

Third, Lemma 4.2 is deferred to [1] with "analogous proof." For the inhomogeneous nonlinearity the energy functional is different enough that a precise reference or a short proof would be better. Minor.

Overall: the result is likely true and the method is worth having. The paper deserves a serious referee. My recommendation is to send it out, with the understanding that the F2/T0 issue must be addressed and the Campos comparison clarified.

Referee Report

2 major / 5 minor

Summary. The paper studies the 2D radial focusing inhomogeneous nonlinear Schrödinger equation i∂_t u + Δu + |x|^{-b}|u|^p u = 0 with 0 < b < 1 and 2 - b < p < ∞. The main result, Theorem 1.1, asserts that for radial initial data below the ground state threshold (in the sense of a mass-energy product), the solution is global and scatters in H^1. The authors propose a proof that avoids the concentration-compactness method, relying instead on a new scattering criterion (Lemma 3.1), radial Sobolev embedding, and a Virial/Morawetz estimate (Proposition 4.1). The paper is an extension of Farah and Guzmán's scattering result from 0 < b < 2/3 to the full range 0 < b < 1.

Significance. If the proof is completed, the result is a meaningful extension of the known scattering theory for the focusing INLS in two dimensions, and it provides an independent route that bypasses concentration compactness. The explicit scattering criterion in Lemma 3.1 and the Morawetz estimate are potentially useful tools for related problems. The paper is self-contained relative to cited well-posedness and interpolation results, and the main argument has no fitted free parameters. However, the proof as written contains a logical gap in the scattering criterion and an omitted proof of a coercivity lemma, so the significance can only be assessed after these issues are resolved.

major comments (2)

[Section 3, proof of Lemma 3.1, estimation of F2 (around (3.14)–(3.15))] There is a logical gap in the treatment of the term F2. The proof chooses T > T0 satisfying (3.15), namely ∫ χ_R |u(T)|^2 < ε^2, but then uses this smallness to control the L∞_t L^2_x norm of χ_R u on the time interval I2 = [T0 - ε^{-θ}, T0]. Since I2 lies to the left of T0 and T is an arbitrary time larger than T0, smallness at T does not propagate to I2 from the derivative bound |d/dt ∫ χ_R |u|^2| ≲ 1/R unless T actually belongs to I2 or the distance from T to I2 is controlled by ε R. As written, the proof is invalid. The gap is fixable by choosing T0 itself from the liminf sequence in (3.10), sufficiently large for the linear-tail estimate, and then setting T = T0 in (3.15); with that choice the smallness at the right endpoint of I2 propagates over I2. The authors should make this choice explicit.
[Section 4, Lemma 4.2] Lemma 4.2 (coercivity on balls) is stated without proof; the text says 'We refer to [1] for an analogous proof.' This lemma is load-bearing: it is used in the proof of Proposition 4.1 to obtain (4.24), which is essential for the Morawetz estimate. Since [1] is described as 'to appear' and the inhomogeneous nonlinearity |x|^{-b} is not treated in [1], the authors should either provide a complete proof or a detailed statement of the analogous argument and explain how it adapts to the present setting. Without this, the proof of Proposition 4.1 is incomplete.

minor comments (5)

[Throughout] There are numerous typographical errors, including 'SCA TTERING' in the title, 'W e' and 'Key W ords' in the abstract, and 'Yangida' in place of 'Yanagida' in reference [19].
[Section 3, Lemma 3.1 statement] The notation M(u0) = E(u0) = E reuses E both as the energy functional and as a generic positive constant; this is confusing. A different symbol, for example L, should be used for the a priori bound in (3.9).
[Section 3, proof of Lemma 3.1, estimation of F2] The expression '|∂t ∫_{I2} χ_R|u|^2 ds|' appears to be a typo; the intended quantity is the absolute value of the time derivative of ∫ χ_R |u|^2 dx. Please correct the notation.
[Section 3, estimation of F1] The Hölder exponents in the display after the dispersive estimate, such as L^{4/(2-p^-)}, contain what appear to be OCR artifacts or typos, especially in the case p > 2 where the exponent 2 - p^- becomes negative. The authors should rewrite this estimate with a valid triple of exponents that covers the full range 2 - b < p < ∞.
[Section 2, definition of admissible pairs] The informal definitions of a^- and a^+ ('a− is a fixed number and slightly smaller than a') should be made precise, since the subsequent estimates rely on these quantities.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the scattering criterion and Morawetz estimates are derived from independent hypotheses, and all load-bearing cited results are external.

full rationale

The paper's central claim, Theorem 1.1, is not obtained by fitting parameters or by defining quantities in terms of the target conclusion. Section 3 proves a new scattering criterion, Lemma 3.1, from explicit assumptions (3.9)-(3.11): a uniform H^1 bound, a small-mass liminf on a ball, and a sublinear space-time bound on the weighted nonlinearity. None of these assumptions is equivalent to scattering or to Theorem 1.1, and the space-time bound (3.11) is itself later derived in Proposition 4.1 via the Virial/Morawetz identity, not assumed. The coercivity Lemma 4.2 is used only to convert the Morawetz identity into the weighted estimate (4.24), and its proof is based on conservation laws and the external Gagliardo-Nirenberg inequality (2.8); the proof is described as analogous to [1], which is an external citation, not a self-citation. Lemmas 2.3 and 2.4, Theorem 2.6, and Proposition 2.5 are cited from Farah, Dinh, Genoud, and other authors, and are independent of the present authors' own results. There is no fitted parameter renamed as a prediction, no ansatz imported from the authors' prior work, and no uniqueness theorem invoked from a self-citation chain. The skeptical objection concerning the role of (3.15) and the interval I2 identifies a possible logical gap in the proof of Lemma 3.1, but that is a correctness issue, not circularity: it concerns the order in which T0, T, and the liminf sequence are chosen, not an input that is equivalent to the output by construction. The derivation chain is therefore self-contained relative to external, standard results, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new free parameters, fitted constants, or invented physical entities. It relies on standard background results from the literature, including well-posedness, Gagliardo-Nirenberg inequalities, Strichartz estimates, and properties of the ground state. The main unproved element is Lemma 4.2, which is deferred to a closely related paper.

assumptions (5)

domain assumption Local and global well-posedness for (1.1) under the ground-state threshold, as cited from Dinh [5] and Farah [8] (Theorem 2.6).
The proof of Theorem 1.1 assumes the solution is global and H^1-bounded; this is provided by the cited well-posedness results, which are not reproved.
domain assumption Sharp Gagliardo-Nirenberg inequality with explicit constant involving the ground state Q (Proposition 2.5, cited from Farah [8]).
Used in Lemma 4.2 to obtain coercivity on large balls. The inequality and the variational characterization of Q are taken from prior literature.
standard math Strichartz estimates (Lemma 2.2) and radial Sobolev embedding (Lemma 2.1).
These standard tools are used throughout the scattering criterion and the Morawetz estimates.
domain assumption Existence and uniqueness of the ground state Q (cited to Genoud, Stuart, and Yanagida).
The threshold in Theorem 1.1 is defined through Q; the paper relies on Q's existence, uniqueness, and the Pohozaev-type identities quoted in Proposition 2.5.
domain assumption Lemma 4.2 (Coercivity on balls) is asserted with proof deferred to Arora-Dodson-Murphy [1].
The Morawetz estimate in Proposition 4.1 depends on this coercivity lemma, whose proof is not reproduced in the paper. The analogous proof in [1] is for the unweighted NLS, so the adaptation to the weighted INLS is not verified.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A remark on the scattering theory for the 2d radial focusing INLS." pith.science (2026). https://pith.science/paper/2NSKW2P3

@misc{pith2026190800743,
  author       = {Pith},
  title        = {Pith review of: A remark on the scattering theory for the 2d radial focusing INLS},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2NSKW2P3}},
  note         = {Machine review of arXiv:1908.00743}
}

abstract

We consider the scattering results of the radial solutions below the ground state to the focusing inhomogeneous nonlinear Schr\"odinger equation $$i\partial_tu+\Delta u +|x|^{-b}|u|^{p}u=0$$ in two dimension, where $0<b<1$ and $2-b<p<\infty$. We use a modified version of Arora-Dodson-Murphy's approach [1] to give a new proof that extends the scattering results of [10] and avoids concentration compactness.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Energy scattering for a class of inhomogeneous nonlinear Schr\"odinger equation in two dimensions
math.AP 2019-08 accept novelty 5.0 of 10

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.

Reference graph

Works this paper leans on

19 extracted references · 16 canonical work pages · cited by 1 Pith paper

[2]

L. Campos. Scattering of radial solutions to the Inhomog eneous nonlinear Schr¨ odinger equation. arXiv:1905.02663, 2019

arXiv 1905
[1]

A. K. Arora, B. Dodson, and J. Murphy. Scattering below gr ound state for the 2d radial nonlinear Schr¨ odinger equation. Proc. Amer. Math. Soc. (to appear)

work page
[3]

V. Dinh. Blowup of H 1 solutions for a class of the focusing inhomogeneous nonline ar Schr¨ odinger equation. Nonlinear Anal. , 174:169–188, 2018

work page 2018
[4]

V. Dinh. Energy scattering for a class of the defocusing i nhomogeneous nonlinear Schr¨ odinger equation. J. Evol. Equ , 19:411–434, 2019

work page 2019
[5]

V. D. Dinh. Scattering theory in a weighted L2 space for a class of the defocusing inhomogeneous nonlinear Schr¨ odinger equation.arXiv:1710.01392, 2017

arXiv 2017
[6]

Dodson and J

B. Dodson and J. Murphy. A new proof of scattering below th e ground state for the 3d radial focusing cubic NLS. Proc. Amer. Math. Soc. , 145(11):4859–4867, 2017

work page 2017
[7]

Duyckaerts, J

T. Duyckaerts, J. Holmer, and S. Roudenko. Scattering fo r the non-radial 3D cubic nonlinear Schr¨ odinger equation. Math. Res. Lett. , 15(6):1233–1250, 2008

work page 2008
[8]

L. G. Farah. Global well-posedness and blow-up on the ene rgy space for the inhomogeneous nonlinear Schr¨ odinger equation.J. Evol. Equ. , 16(1):193–208, 2016

work page 2016

Show all 19 references

[9]

L. G. Farah and C. M. Guzm´ an. Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schr¨ odinger equation.J. Diﬀerential Equations , 262(8):4175–4231, 2017

2017
[10]

L. G. Farah and C. M. Guzm´ an. Scattering for the radial f ocusing INLS equation in higher dimensions. Bull Braz Math Soc, New Series , 2019

2019
[11]

F. Genoud. Th´ eorie de bifuration et de stabilit´ e pour une ´equation de schr¨ odinger avec une non-lin´earit´e compacte. Ph.D. Thesis , 2008

2008
[12]

F. Genoud. Bifurcation and stability of travelling wav es in self-focusing planar waveguides. Adv. Nonlinear Stud., 10(2):357–400, 2010

2010
[13]

F. Genoud. A uniqueness result for ∆ u − λu + V (|x|)up = 0 on R2. Adv. Nonlinear Stud. , 11(3):483–491, 2011

2011
[14]

Genoud and C

F. Genoud and C. A. Stuart. Schr¨ odinger equations with a spatially decaying nonlinearity: existence and stability of standing waves. Discrete Contin. Dyn. Syst. , 21(1):137, 2008

2008
[15]

C. M. Guzm´ an. On well posedness for the inhomogeneous n onlinear Schr¨ odinger equation.Nonlinear Anal. Real World Appl. , 37:249–286, 2017

2017
[16]

Holmer and S

J. Holmer and S. Roudenko. A sharp condition for scatter ing of the radial 3D cubic nonlinear Schr¨ odinger equation. Comm. Math. Phys. , 282(2):435–467, 2008

2008
[17]

Keel and T

M. Keel and T. Tao. Endpoint Strichartz estimates. Amer. J. Math. , 120(5):955–980, 1998

1998
[18]

T. Tao. On the asymptotic behavior of large radial data f or a focusing non-linear Schr¨ odinger equation. Dyn. Partial Diﬀer. Equ. , 1(1):1–48, 2004

2004
[19]

Yanagida

E. Yanagida. Uniqueness of positive radial solutions o f δu + g(r)u + h(r)up = 0 in ρn. Arch. Ration. Mech. Anal., 115(3):257–274, 1991. The Graduate School of China Academy of Engineering Physics , P. O. Box 2101, Beijing, China, 100088 E-mail address : xcbsph@163.com Beijing...

1991

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [2]

L. Campos. Scattering of radial solutions to the Inhomog eneous nonlinear Schr¨ odinger equation. arXiv:1905.02663, 2019

arXiv 1905

[2] [1]

A. K. Arora, B. Dodson, and J. Murphy. Scattering below gr ound state for the 2d radial nonlinear Schr¨ odinger equation. Proc. Amer. Math. Soc. (to appear)

work page

[3] [3]

V. Dinh. Blowup of H 1 solutions for a class of the focusing inhomogeneous nonline ar Schr¨ odinger equation. Nonlinear Anal. , 174:169–188, 2018

work page 2018

[4] [4]

V. Dinh. Energy scattering for a class of the defocusing i nhomogeneous nonlinear Schr¨ odinger equation. J. Evol. Equ , 19:411–434, 2019

work page 2019

[5] [5]

V. D. Dinh. Scattering theory in a weighted L2 space for a class of the defocusing inhomogeneous nonlinear Schr¨ odinger equation.arXiv:1710.01392, 2017

arXiv 2017

[6] [6]

Dodson and J

B. Dodson and J. Murphy. A new proof of scattering below th e ground state for the 3d radial focusing cubic NLS. Proc. Amer. Math. Soc. , 145(11):4859–4867, 2017

work page 2017

[7] [7]

Duyckaerts, J

T. Duyckaerts, J. Holmer, and S. Roudenko. Scattering fo r the non-radial 3D cubic nonlinear Schr¨ odinger equation. Math. Res. Lett. , 15(6):1233–1250, 2008

work page 2008

[8] [8]

L. G. Farah. Global well-posedness and blow-up on the ene rgy space for the inhomogeneous nonlinear Schr¨ odinger equation.J. Evol. Equ. , 16(1):193–208, 2016

work page 2016