Maximum Decay Rate for the Nonlinear Schr\"odinger Equation

TL;DR

This paper establishes that solutions to the nonlinear Schrödinger equation cannot decay faster than free solutions, given initial data in certain Sobolev and weighted spaces, highlighting fundamental decay limitations.

Contribution

It proves a maximum decay rate for nonlinear Schrödinger solutions, extending understanding of decay behavior in various initial data spaces.

Findings

No nontrivial solution decays faster than free Schrödinger solutions.

Decay rate bounds depend on initial data spaces.

Results apply to a broad class of nonlinearities with subcritical exponents.

Abstract

In this paper, we consider global solutions for the following nonlinear Schr\"odinger equation $iu_t+\Delta u+\lambda|u|^\alpha u=0,$ in $\R^N,$ with $\lambda\in\R$ 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 Schr\"odinger 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.