On Decaying Properties of Nonlinear Schrödinger Equations

.In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing nonlinear Schrödinger equation with various (deterministic and random) initial data. We show that nonlinear solutions enjoy the same decay rate as the linear ones. The regularity assumption on the initial data is much lower than in previous results (see [C. Fan and Z. Zhao, Discrete Contin. Dyn. Syst., 41 (2021), pp. 3973–3984] and the references therein), and, moreover, we quantify the decay, which is another novelty of this work. Furthermore, we show that the (physical) randomization of the initial data can be used to replace the \(L^1\)-data assumption (see [C. Fan and Z. Zhao, Proc. Amer. Math. Soc., 151 (2023), pp. 2527–2542] for the necessity of the \(L^1\)-data assumption).