. A P ] 1 3 N ov 2 00 6 A ( CONCENTRATION - ) COMPACT ATTRACTOR FOR HIGH - DIMENSIONAL NON - LINEAR SCHRÖDINGER EQUATIONS
نویسنده
چکیده
We study the asymptotic behavior of large data solutions to Schrödinger equations iu t + ∆u = F (u) in R d , assuming globally bounded H 1 x (R d) norm (i.e. no blowup in the energy space), in high dimensions d ≥ 5 and with nonlinearity which is energy-subcritical and mass-supercritical. In the spherically symmetric case, we show that as t → +∞, these solutions split into a radiation term that evolves according to the linear Schrödinger equation, and a remainder which converges in H 1 x (R d) to a compact attractor, which consists of the union of spherically symmetric almost periodic orbits of the NLS flow in H 1 x (R d). This is despite the total lack of any dissipation in the equation. This statement can be viewed as weak form of the " soliton resolution conjecture ". We also obtain a more complicated analogue of this result for the non-spherically-symmetric case. As a corollary we obtain the " petite conjecture " of Soffer in the high dimensional non-critical case.
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