, Volume 345, Issue 1, pp 133-158
Date: 19 Mar 2009

Semilinear Schrödinger flows on hyperbolic spaces: scattering in H 1

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Abstract

We prove global well-posedness and scattering in H 1 for the defocusing nonlinear Schrödinger equations $$\left\{\begin{array}{ll}(i\partial_t+\Delta_g)u=u|u|^{2\sigma};\\u(0)=\phi,\end{array}\right.$$ 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.

A. D. Ionescu was supported in part by a Packard Fellowship. G. Staffilani was supported in part by NSF Grant DMS 0602678