Semilinear Schrödinger Flows on Hyperbolic Spaces: Scattering in H^1

Details Semilinear Schrödinger Flows on Hyperbolic Spaces: Scattering in H^1 Semilinear Schrödinger Flows on Hyperbolic Spaces: Scattering in H^1

2008-01-18

arxiv.org/abs/0801.2957v1

Mathematics

Analysis of PDEs

Scientific paper

We prove global well-posedness and scattering in $H^1$ for the defocusing nonlinear Schr\"{o}dinger equations \begin{equation*} \begin{cases} &(i\partial_t+\Delta_\g)u=u|u|^{2\sigma}; &u(0)=\phi, \end{cases} \end{equation*} on the hyperbolic spaces $\H^d$, $d\geq 2$, for exponents $\sigma\in(0,2/(d-2))$. The main unexpected conclusion is scattering to linear solutions in the case of small exponents $\sigma$; 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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