Maximum Decay Rate for the Nonlinear Schrödinger Equation

Abstract

In this paper, we consider global solutions for the following nonlinear Schrödinger equation $i{u}_t+\Delta u+\lambda |u{|}^{\alpha }u=0,$ in ${\mathbb{R}}^N,$ with $\lambda \in \mathbb{R}$ 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 $H^1({\mathbb{R}}^N)\cap L^2(|x{|}^2;dx),$ in the energy space, namely $H^1({\mathbb{R}}^N),$ or in $L^2({\mathbb{R}}^N),$ according to the different cases.