Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions of the Nonlinear Schrödinger Equation" the following property. Given the problem \begin{equation} \left\{ \begin{array}{rl} iu_t + \Delta u+uu^{\alpha}=0\\ u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\ \end{array}\right. \end{equation} if $u$ is a global solution of the problem above such that $u\in L^a((0,\infty),L^r(\mathbb{R}^N))$, then exists a scattering state $u^+\in H^1(\mathbb{R}^N)$ such that \begin{equation} \u(t)e^{it\Delta}u^+\_{H^1}\overset{t\rightarrow \infty}{\longrightarrow} 0, \end{equation} where $r=\alpha +2$, $a=\frac{2(\alpha+2)\alpha}{4(N2)\alpha}$ and $\frac4N<\alpha<\frac{4}{N2}$. My question is: if $u$ scatters to some $u^+$ can I say that $u\in L^a((0,\infty),L^r(\mathbb{R}^N))$?
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3$\begingroup$ You will need some minimal hypotheses on u, e.g. $C^0_t H^1_x$, and to specify in what sense $u$ is to be a solution (e.g. a strong $H^1$ solution). Given such hypotheses, the existing results on asymptotic completeness and unconditional uniqueness for NLS should apply (albeit with the caveat that the unconditional uniqueness results have to be used starting from time $t=+\infty$ rather than from $t=0$). $\endgroup$– Terry TaoMar 13, 2014 at 16:54
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