Abstract.
We study the long-time behavior of radial solutions to nonlinear Schrödinger equations on hyperbolic space. We show that the usual distinction between short-range and long-range nonlinearity is modified: the geometry of the hyperbolic space makes every power-like nonlinearity short range. The proofs rely on weighted Strichartz estimates, which imply Strichartz estimates for a broader family of admissible pairs, and on Morawetz-type inequalities. The latter are established without symmetry assumptions.
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V.B. is partially supported by the ANR project “Étude qualitative des E.D.P.”. R.C. acknowledges support by the ANR project SCASEN. G.S. is partially supported by NSF Grant 0602678.
Received: July 2006, Revision: April 2007, Accepted: April 2007
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Banica, V., Carles, R. & Staffilani, G. Scattering Theory for Radial Nonlinear Schrödinger Equations on Hyperbolic Space. GAFA Geom. funct. anal. 18, 367–399 (2008). https://doi.org/10.1007/s00039-008-0663-x
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DOI: https://doi.org/10.1007/s00039-008-0663-x
Keywords and phrases:
- Nonlinear Schrödinger equations on manifolds
- asymptotic behavior
- Strichartz estimates
- Morawetz estimates