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

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.