Decay estimates for nonlinear Schrödinger equation with the inverse-square potential

Abstract

In this paper, we study the dispersive decay estimates for solution to the $3D$ energy-critical nonlinear Schrödinger equation with an inverse-square operator $L_a$ where the operator is denoted by $L_a:=-\Delta +\frac{a}{|x{|}^2}$ with the constant $a\ge 0$. Inspired by the work of [20], [23], we first establish that the solutions exhibit ${\dot{H}}^1\left(R^3\right)$ uniform regularity, derive the Lorentz-Strichartz estimates, and then obtain the desired decay estimates using the bootstrap argument. The key ingredients of our approach include the equivalence of Sobolev norms and the fractional product rule.