2,151 research outputs found
Near Sharp Strichartz estimates with loss in the presence of degenerate hyperbolic trapping
We consider an -dimensional spherically symmetric, asymptotically
Euclidean manifold with two ends and a codimension 1 trapped set which is
degenerately hyperbolic. By separating variables and constructing a
semiclassical parametrix for a time scale polynomially beyond Ehrenfest time,
we show that solutions to the linear Schr\"odiner equation with initial
conditions localized on a spherical harmonic satisfy Strichartz estimates with
a loss depending only on the dimension and independent of the degeneracy.
The Strichartz estimates are sharp up to an arbitrary loss. This is
in contrast to \cite{ChWu-lsm}, where it is shown that solutions satisfy a
sharp local smoothing estimate with loss depending only on the degeneracy of
the trapped set, independent of the dimension
Existence and stability of solitons for the nonlinear Schr\"odinger equation on hyperbolic space
We study the existence and stability of ground state solutions or solitons to
a nonlinear stationary equation on hyperbolic space. The method of
concentration compactness applies and shows that the results correlate strongly
to those of Euclidean space.Comment: New: As noted in Banica-Duyckaerts (arXiv:1411.0846), Section 5
should read that for sufficiently large mass, sub-critical problems can be
solved via energy minimization for all d \geq 2 and as a result
Cazenave-Lions results can be applied in Section 6 with the same restriction.
These requirements were addressed by the subsequent work with Metcalfe and
Taylor in arXiv:1203.361
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