Dispersive decay for the energy-critical nonlinear Schr\"odinger equation

TL;DR

This paper proves that solutions to the energy-critical nonlinear Schrödinger equation in 3 and 4 dimensions exhibit dispersive decay over time, applicable to both initial-value and final-state problems.

Contribution

It establishes pointwise-in-time dispersive decay for solutions in energy-critical cases in dimensions 3 and 4, advancing understanding of long-term behavior.

Findings

Proves dispersive decay in 3 and 4 dimensions

Applies to both initial-value and final-state problems

Enhances understanding of energy-critical nonlinear Schrödinger equations

Abstract

We prove pointwise-in-time dispersive decay for solutions to the energy-critical nonlinear Schr\"odinger equation in spatial dimensions $d = 3,4$ for both the initial-value and final-state problems.