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

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dc.creatorTao, Terence
dc.date2008-05-11
dc.date2008-05-28
dc.date.accessioned2026-07-07T09:41:02Z
dc.date.available2026-07-07T09:41:02Z
dc.descriptionWe 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 Schrödinger equations $i u_t + Δu = |u|^{p-1} u + Vu$ in $\R^d$, where $V \in C^\infty_0(\R^d)$ is a real potential (which could contain bound states), and $1+\frac{4}{d} < p < 1+\frac{4}{d-2}$ is an exponent which 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.
dc.description19 pages, no figures, submitted, Dynamics of PDE. Referee corrections incorporated
dc.identifierhttps://arxiv.org/abs/0805.1544
dc.identifierhttp://arxiv.org/abs/0805.1544
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/161689
dc.subjectAnalysis of PDEs
dc.subject35Q55
dc.titleA global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential
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