A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential

Abstract

We study the asymptotic behavior of large data solutions in the energy space $H:=H^1({\mathbb{R}}^d)$ in very high dimension $d\ge 11$ to defocusing Schrödinger equations $i{u}_t+\Delta u=|u{|}^{p-1}u+Vu$ in ${\mathbb{R}}^d$, where $V\in C_0^{\infty }({\mathbb{R}}^d)$ is a real potential (which could contain bound states), and $1+\frac{4}{d}<p<1+\frac{4}{d-2}$ is an exponent 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 Schrödinger equation, and a remainder which converges in $H$ to a compact attractor $K$, which consists of the union of spherically symmetric almost periodic orbits of the NLS flow in $H$. The main novelty of this result is that $K$ is a \emph{global} attractor, being independent of the initial energy of the initial data; in particular, no matter how large the initial data is, all but a bounded amount of energy is radiated away in the limit.

19 pages, no figures, submitted, Dynamics of PDE. Referee corrections incorporated