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

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.