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Global well-posedness and scattering of energy-critical defocusing nonlinear Schrödinger equation on high-dimensional hyperbolic spaces
Abstract
We prove global well-posedness and scattering for the energy-critical defocusing nonlinear Schrödinger equation i∂t u + Δu = |u|4/(d−2) u, u(0) = u0, on hyperbolic space ℍd(d ≥ 4). We mainly utilize the concentration-compactness/rigidity method. We establish a linear profile decomposition on high-dimensional hyperbolic spaces and prove that the nonlinear profile on hyperbolic spaces can be approximated by the solutions of the energy-critical defocusing nonlinear Schrödinger equation on Euclidean spaces, then obtaining the global well-posedness and scattering of the energy-critical defocusing nonlinear Schrödinger equation on the high-dimensional hyperbolic space.
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BibTeX
@misc{zhang2026global,
title = {{Global well-posedness and scattering of energy-critical defocusing nonlinear Schrödinger equation on high-dimensional hyperbolic spaces}},
author = {Shunqin Zhang and Jiqiang Zheng and Yunrui Zheng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37077},
eprint = {2610.37077},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37077},
}
RIS
TY - UNPB
TI - Global well-posedness and scattering of energy-critical defocusing nonlinear Schrödinger equation on high-dimensional hyperbolic spaces
AU - Shunqin Zhang
AU - Jiqiang Zheng
AU - Yunrui Zheng
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37077
UR - https://arxiv.org/abs/2610.37077
AB - We prove global well-posedness and scattering for the energy-critical defocusing nonlinear Schrödinger equation i∂_t u + Δu = |u|^4/(d−2) u, u(0) = u_0, on hyperbolic space ℍ^d(d ≥ 4). We mainly utilize the concentration-compactness/rigidity method. We establish a linear profile decomposition on high-dimensional hyperbolic spaces and prove that the nonlinear profile on hyperbolic spaces can be approximated by the solutions of the energy-critical defocusing nonlinear Schrödinger equation on Euclidean spaces, then obtaining the global well-posedness and scattering of the energy-critical defocusing nonlinear Schrödinger equation on the high-dimensional hyperbolic space.
ER -