Misc,
A global compact attractor for high-dimensional defocusing non-linear Schr\ödinger equations with potential
(2008)
Abstract
We study the asymptotic behavior of large data solutions in the energy space $H := H^1(\R^d)$ in very high dimension $d 11$ to defocusing Schrödinger equations $i u_t + \Delta u = |u|^p-1 u + Vu$ in $\R^d$, where $V C^ınfty_0(\R^d)$ is a real potential (which could contain bound states), and $1+4d < p < 1+4d-2$ is an exponent which is energy-subcritical and mass-supercritical. In the spherically symmetric case, we show that as $t +ınfty$, 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 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.
