IUMJ

Title: Spatial decay and time-asymptotic profiles for solutions of Schrodinger equations

Authors: Thierry Cazenave and Fred B. Weissler

Issue: Volume 55 (2006), Issue 1, 75-118

Abstract:

In this paper, we study the relationship between the long time behavior of a solution $e^{it\Delta}\varphi$ of the Schroedinger equation on $\mathbb{R}^N$ and the asymptotic behavior as $|x| \to \infty$ of its initial value $\varphi$. Under appropriate hypotheses on $\varphi$ we show that, for a fixed $0 < \sigma < N$, if the sequence of dilations $\lambda_n^{\sigma} \varphi (\lambda_n \cdot)$ converges in $\mathcal{S}'(\mathbb{R}^N)$ to $\psi(\cdot)$ as $\lambda_n \to \infty$, then the rescaled solution $t^{\sigma/2}e^{it\Delta} \varphi (\cdot\sqrt t)$ converges in $L^r(\mathbb{R}^N)$, for $r$ sufficiently large, to $e^{i\Delta}\psi$ along the subsequence $t_n = \lambda_n^2$. Moreover, we show there exists an initial value $\varphi$ (in $H^{\infty} (\mathbb{R}^N)$ if $\sigma > N/2$) such that the set of all possible $\psi$ attainable in this fashion is a closed ball $B$ of an infinite dimensional Banach space. The resulting "universal" solution is therefore asymptotically close along appropriate subsequences to all solutions with initial values in $B$. Furthermore, $e^{i\Delta}$ followed by an appropriate dilation generates a chaotic discrete dynamical system on a compact subset of $L^r(\mathbb{R}^N)$. Finally, we prove analogous results for the nonlinear Schroedinger equation.