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.
Description
[0805.1544] A global compact attractor for high-dimensional defocusing non-linear
Schr\"odinger equations with potential
%0 Generic
%1 tao-2008
%A Tao, Terence
%D 2008
%K NLS attractor defocusing imported
%T A global compact attractor for high-dimensional defocusing non-linear Schr\ödinger equations with potential
%U http://www.citebase.org/abstract?id=oai:arXiv.org:0805.1544
%X 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.
@misc{tao-2008,
abstract = { We study the asymptotic behavior of large data solutions in the energy space $H := H^1(\R^d)$ in very high dimension $d \geq 11$ to defocusing Schr\"odinger equations $i u_t + \Delta u = |u|^{p-1} u + Vu$ 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 Schr\"odinger 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.},
added-at = {2008-05-21T08:07:23.000+0200},
author = {Tao, Terence},
biburl = {https://www.bibsonomy.org/bibtex/22c3cc7be73f65c85cea2755ad82bbdd1/ralfwit},
description = {[0805.1544] A global compact attractor for high-dimensional defocusing non-linear
Schr\"odinger equations with potential},
interhash = {bb96111b0b0c553ac85b55a330be0186},
intrahash = {2c3cc7be73f65c85cea2755ad82bbdd1},
keywords = {NLS attractor defocusing imported},
timestamp = {2008-05-21T08:07:23.000+0200},
title = {A global compact attractor for high-dimensional defocusing non-linear Schr\\"odinger equations with potential},
url = {http://www.citebase.org/abstract?id=oai:arXiv.org:0805.1544},
year = 2008
}