Maximum Decay Rate for Finite-Energy Solutions of Nonlinear Schr\"odinger Equations

TL;DR

This paper establishes explicit lower bounds on the decay rates over time for all nontrivial finite-energy solutions of nonlinear Schrödinger equations across various domains, extending previous results to broader nonlinearities.

Contribution

It provides new explicit decay rate bounds applicable to a wide class of nonlinear Schrödinger equations in any domain, generalizing earlier findings.

Findings

Explicit time lower bounds for solutions

Applicable to all nontrivial solutions in energy space

Valid for broad class of nonlinearities

Abstract

We give explicit time lower bounds in the Lebesgue spaces for all nontrivial solutions of nonlinear Schr\"odinger equations bounded in the energy space. The result applies for these equations set in any domain of $\R^N,$ including the whole space. This also holds for a large class of nonlinearities, thereby extending the results obtained by Hayashi and Ozawa in \cite{MR91d:35035} and by the author in \cite{beg3}.