Dispersive decay for the Inter-critical nonlinear Schr\"{o}dinger equation in $\mathbb{R}^3$

TL;DR

This paper establishes uniform decay estimates for solutions to the inter-critical nonlinear Schrödinger equation in three dimensions, extending previous results to initial data in the critical Sobolev space.

Contribution

It provides new decay estimates for the NLS in the mass-supercritical and energy-subcritical regime with initial data in the critical Sobolev space.

Findings

Uniform decay estimates for solutions in $ ext{H}^{s_c}$

Extension of previous decay results to broader initial data class

Insights into long-time behavior of inter-critical NLS solutions

Abstract

This paper investigates the Cauchy problem for the nonlinear Schr\"odinger equation (NLS) in the mass-supercritical and energy-subcritical regime within three spatial dimensions. For initial data in the critical homogeneous Sobolev space $\dot{H}^{s_c}(\mathbb{R}^3)$ (where $s_c = \frac{5}{6}$), we get a uniform decay estimate for the long-time dynamics of solutions, which extends the previous results.