MR2829549
Daoyuan FANG, Jian XIE and Thierry CAZENAVE
Daoyuan FANG, Jian XIE and Thierry CAZENAVE
Multiscale Asymptotic Behavior of the Schrödinger Equation
Funkcialaj Ekvacioj. Serio Internacia
54
2011
69--94
http://fe.math.kobe-u.ac.jp/FE/FullPapers/54-1/54_69.pdf
http://www.ams.org/mathscinet-getitem?mr=MR2829549
In this paper, we construct solutions $e^{it\Delta}\phi$ of the Schrödinger equation on ${\bf R}^N$ which have nontrivial asymptotic properties simultaneously on different time and space scales. More precisely, given $\mu\in (0,N)$ and $\beta \ge 1/2$ we consider the set $\omega_{\mu,r}^\beta (\phi)$ of limit points in $L^r({\bf R}^N)$ as $t\to \infty $ of $t^{\mu/2}[e^{it\Delta}\phi](\cdot t^\beta)$. We show in particular that, given $0<\nu<N$ and an arbitrary countable set $S\subset(\nu,N)$, there exists an initial value $\phi$ such that $\omega_{\mu,r}^\beta (\phi)=L^r({\bf R}^N)$ for all $\mu \in (0,N)$ and $\beta \ge 1/2$ such that $\mu /2\beta \in S$, and all sufficiently large $r$. We also establish a result of a similar nature for a nonlinear Schrödinger equation.
Schrödinger's equation, Asymptotic behavior, $\omega$-limit set.
Primary: 35Q55, Secondary: 35B40.
54-69
2011
Multiscale Asymptotic Behavior of the Schrödinger Equation
Daoyuan FANG, Jian XIE and Thierry CAZENAVE
Daoyuan FANG, Jian XIE and Thierry CAZENAVE
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