Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions of the Nonlinear SchrĂ¶dinger 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 $u\in L^a((0,\infty),L^r(\mathbb{R}^N))$?
