PreprintarXiv licencearXiv
Global theory for NLS in the weighted spaces II: final data problem
Abstract
In this paper, we study the final data problem for the defocusing nonlinear Schrödinger equation (NLS) i∂t u + 1/2Δu = |u|p u in weighted spaces Σ̇s(ℝd):=Lx2(ℝd;|x|2sd x). Let sc=d/2−2/p. Under the scaling of the NLS, Σ̇−sc(ℝd) is the scaling-critical weighted space. In three dimensions, within the range of exponents considered in this paper, we show that the final data problem is locally well-posed for s≥ −sc and ill-posed for s<−sc. This is opposite to the corresponding initial data problem, which is locally well-posed for s≤ −sc and ill-posed for s>−sc. For the three-dimensional quadratic equation, we further obtain a unique global solution for every radial final datum in the critical space Σ̇1/2(ℝ3).
Related
- Global theory for NLS in the weighted spaces I: finite pseudo conformal energy
- The Reciprocal Problem on Weighted Bergman Spaces
- Localization and weighted theory on the Bergman spaces
- IDA symbols and Schatten--Lorentz theory for Toeplitz operators on weighted Fock spaces
- The Stokes Operator on Exterior Domains in Homogeneous Weighted Function Spaces:From Weak Theory to ℋ∞-calculus to Fractional Domains
BibTeX
@misc{guo2026global,
title = {{Global theory for NLS in the weighted spaces II: final data problem}},
author = {Yujin Guo and Jia Shen and Yifei Wu and Changping Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37785},
eprint = {2609.37785},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37785},
}
RIS
TY - UNPB
TI - Global theory for NLS in the weighted spaces II: final data problem
AU - Yujin Guo
AU - Jia Shen
AU - Yifei Wu
AU - Changping Yang
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37785
UR - https://arxiv.org/abs/2609.37785
AB - In this paper, we study the final data problem for the defocusing nonlinear Schrödinger equation (NLS) i∂_t u + 1/2Δu = |u|^p u in weighted spaces Σ̇^s(ℝ^d):=L_x^2(ℝ^d;|x|^2sd x). Let s_c=d/2−2/p. Under the scaling of the NLS, Σ̇^−s_c(ℝ^d) is the scaling-critical weighted space. In three dimensions, within the range of exponents considered in this paper, we show that the final data problem is locally well-posed for s≥ −s_c and ill-posed for s<−s_c. This is opposite to the corresponding initial data problem, which is locally well-posed for s≤ −s_c and ill-posed for s>−s_c. For the three-dimensional quadratic equation, we further obtain a unique global solution for every radial final datum in the critical space Σ̇^1/2(ℝ^3).
ER -