Mathematics – Analysis of PDEs
Scientific paper
2006-11-13
Dynamics of PDE 4 (2007), 1-53
Mathematics
Analysis of PDEs
60 pages. An error in the proof of the perturbation lemma (pointed out by Tristan Roy) is now fixed
Scientific paper
We study the asymptotic behavior of large data solutions to Schr\"odinger equations $i u_t + \Delta u = F(u)$ in $\R^d$, assuming globally bounded $H^1_x(\R^d)$ norm (i.e. no blowup in the energy space), in high dimensions $d \geq 5$ and with nonlinearity which is energy-subcritical and mass-supercritical. In the spherically symmetric case, we show that as $t \to +\infty$, these solutions split into a radiation term that evolves according to the linear Schr\"odinger 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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