Dispersive decay for the Inter-critical nonlinear Schrödinger equation in ${\mathbb{R}}^3$

This paper investigates the Cauchy problem for the nonlinear Schrödinger 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.