Global existence, scattering and blow-up for the focusing NLS on the hyperbolic space

2014/11/04 by Valeria Banica, Banica, Valeria, Thomas Duyckaerts +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #math.AP

paper · pdf · doi:10.48550/arxiv.1411.0846

Small corrections in the references

openalex publication_date 2014/11/04 · arxiv created 2014/11/14 · arxiv updated 2014/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove global well-posedness, scattering and blow-up results for energy-subcritical focusing nonlinear Schrödinger equations on the hyperbolic space. We show in particular the existence of a critical element for scattering for all energy-subcritical power nonlinearities. For mass-supercritical nonlinearity, we show a scattering vs blow-up dichotomy for radial solutions of the equation in low dimension, below natural mass and energy thresholds given by the ground states of the equation. The proofs are based on trapping by mass and energy, compactness and rigidity, and are similar to the ones on the Euclidean space, with a new argument, based on generalized Pohozaev identities, to obtain appropriate monotonicity formulas.

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