English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We study the asymptotic behavior of large data solutions in the energy space$H := H^1(\R^d)$ in very high dimension $d \geq 11$ to defocusing Schr\"odingerequations $i u_t + \Delta u = |u|^{p-1} u + Vu$ in $\R^d$, where $V \inC^\infty_0(\R^d)$ is a real potential (which could contain bound states), and$1+\frac{4}{d} < p < 1+\frac{4}{d-2}$ is an exponent which isenergy-subcritical and mass-supercritical. In the spherically symmetric case,we show that as $t \to +\infty$, these solutions split into a radiation termthat evolves according to the linear Schr\"odinger equation, and a remainderwhich converges in $H$ to a compact attractor $K$, which consists of the unionof spherically symmetric almost periodic orbits of the NLS flow in $H$. Themain novelty of this result is that $K$ is a \emph{global} attractor, beingindependent of the initial energy of the initial data; in particular, no matterhow large the initial data is, all but a bounded amount of energy is radiatedaway in the limit.

A global compact attractor for high-dimensional defocusing non-linear Schr\"odinger equations with potential