Bégout, Pascal (2004) Maximum Decay Rate for the Nonlinear Schrödinger Equation. NoDEA : Nonlinear Differential Equations and Applications, 11 (4). pp. 451-467.

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Abstract

In this paper, we consider global solutions for the following nonlinear Schrödinger equation $iu_t+\Delta u+\lambda|u|^\alpha u=0,$ in $\R^N,$ with
$\lambda\in\R$ and $0\le\alpha<\frac{4}{N-2}$ $(0\le\alpha<\infty$ if $N=1).$ We show that no nontrivial solution can decay faster than the solutions of the free Schrödinger equation, provided that $u(0)$ lies in the weighted Sobolev space $H^1(\R^N)\cap L^2(|x|^2;dx),$ in the energy space, namely $H^1(\R^N),$ or in $L^2(\R^N),$ according to the different cases.