Maximum Decay Rate for the Nonlinear Schrödinger Equation

In this paper, we consider global solutions for the following nonlinear Schrödinger equation $iu_t+Δu+λ|u|^αu=0,$ in $\R^N,$ with $λ\in\R$ and $0\leα<\frac{4}{N-2}$ $(0\leα<\infty$ 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.

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