Abstract
We prove global well-posedness and scattering in H 1 for the defocusing nonlinear Schrödinger equations
on the hyperbolic spaces \({\mathbb{H}^d}\), d ≥ 2, for exponents \({\sigma \in (0, 2/(d-2))}\). The main unexpected conclusion is scattering to linear solutions in the case of small exponents σ; for comparison, on Euclidean spaces scattering in H 1 is not known for any exponent \({\sigma \in (1/d, 2/d]}\) and is known to fail for \({\sigma \in (0, 1/d]}\). Our main ingredients are certain noneuclidean global in time Strichartz estimates and noneuclidean Morawetz inequalities.
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A. D. Ionescu was supported in part by a Packard Fellowship. G. Staffilani was supported in part by NSF Grant DMS 0602678
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Ionescu, A.D., Staffilani, G. Semilinear Schrödinger flows on hyperbolic spaces: scattering in H 1 . Math. Ann. 345, 133–158 (2009). https://doi.org/10.1007/s00208-009-0344-6
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DOI: https://doi.org/10.1007/s00208-009-0344-6