Global well-posedness and scattering for defocusing energy-critical inhomogeneous NLS in dimensions $d\ge 3$
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2026
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| _version_ | 1866918453153103872 |
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| author | Yang, Bo Zhang, Lei Liu, Bin |
| author_facet | Yang, Bo Zhang, Lei Liu, Bin |
| contents | We study the defocusing energy-critical inhomogeneous nonlinear Schrödinger equation \[
i\partial_tu+Δu=|x|^{-b}|u|^{\frac{4-2b}{d-2}}u,
\qquad (t,x)\in\R\times\R^d, \] with initial data $u_0\in\dot H_x^1(\R^d)$, where $d\ge 3$ and $0<b<\min\{2,\frac d2\}$. We prove global well-posedness and scattering for arbitrary non-radial data. The main difficulties are that, when $d\ge 6$, the derivative of the critical nonlinearity is only Hölder continuous, so the short-time perturbation argument cannot be closed in $\dot S^1$, and that the singular coefficient $|x|^{-b}$ breaks translation symmetry. To handle these issues, we exploit the weak-space structure $|x|^{-b}\in L^{\frac{d}{b},\infty}(\R^d)$, introduce exotic Strichartz norms, and prove a long-time stability theorem for the general energy-critical inhomogeneous nonlinear Schrödinger equation. We also show that profiles escaping to spatial infinity are asymptotically linear because of the decay of $|x|^{-b}$. Consequently, almost periodic solutions are compact modulo scaling only, with neither spatial nor frequency center parameters. Combined with the concentration--compactness argument of Kenig--Merle [\emph{Invent. Math.} \textbf{166} (2006), 645--675], this yields the main theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_16856 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global well-posedness and scattering for defocusing energy-critical inhomogeneous NLS in dimensions $d\ge 3$ Yang, Bo Zhang, Lei Liu, Bin Analysis of PDEs We study the defocusing energy-critical inhomogeneous nonlinear Schrödinger equation \[ i\partial_tu+Δu=|x|^{-b}|u|^{\frac{4-2b}{d-2}}u, \qquad (t,x)\in\R\times\R^d, \] with initial data $u_0\in\dot H_x^1(\R^d)$, where $d\ge 3$ and $0<b<\min\{2,\frac d2\}$. We prove global well-posedness and scattering for arbitrary non-radial data. The main difficulties are that, when $d\ge 6$, the derivative of the critical nonlinearity is only Hölder continuous, so the short-time perturbation argument cannot be closed in $\dot S^1$, and that the singular coefficient $|x|^{-b}$ breaks translation symmetry. To handle these issues, we exploit the weak-space structure $|x|^{-b}\in L^{\frac{d}{b},\infty}(\R^d)$, introduce exotic Strichartz norms, and prove a long-time stability theorem for the general energy-critical inhomogeneous nonlinear Schrödinger equation. We also show that profiles escaping to spatial infinity are asymptotically linear because of the decay of $|x|^{-b}$. Consequently, almost periodic solutions are compact modulo scaling only, with neither spatial nor frequency center parameters. Combined with the concentration--compactness argument of Kenig--Merle [\emph{Invent. Math.} \textbf{166} (2006), 645--675], this yields the main theorem. |
| title | Global well-posedness and scattering for defocusing energy-critical inhomogeneous NLS in dimensions $d\ge 3$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.16856 |