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# A note on decay property of nonlinear Schr\"odinger equations

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### Abstract

In this note, we show the existence of a special solution $$u$$ to defocusing cubic NLS in $$3d$$, which lives in $$H^{s}$$ for all $$s>0$$, but scatters to a linear solution in a very slow way. We prove for this $$u$$, for all $$\epsilon>0$$, one has $$\sup_{t>0}t^{\epsilon}\|u(t)-e^{it\Delta}u^{+}\|_{\dot{H}^{1/2}}=\infty$$. Note that such a slow asymptotic convergence is impossible if one further pose the initial data of $$u(0)$$ be in $$L^{1}$$. We expect that similar construction holds for the other NLS models.

### Author and article information

###### Journal
14 March 2022
###### Article
2203.06896
fc8d6612-25a6-46f7-ba0e-e8e5f487030f