On the inhomogeneous NLS with inverse-square potential

TL;DR

This paper investigates the inhomogeneous nonlinear Schrödinger equation with inverse-square potential, establishing conditions for global existence, blow-up, and scattering, and constructing wave operators in the energy space.

Contribution

It provides new criteria for global solutions, blow-up, and scattering in the inhomogeneous NLS with inverse-square potential, extending previous results to more general cases.

Findings

Established sufficient conditions for global existence and blow-up.

Proved local and global well-posedness using Strichartz estimates.

Constructed wave operators in the energy space.

Abstract

We consider the inhomogeneous nonlinear Schr\"odinger equation with inverse-square potential in $\mathbb{R}^N$ $$ i u_t + \mathcal{L}_a u+\lambda |x|^{-b}|u|^\alpha u = 0,\;\;\mathcal{L}_a=\Delta -\frac{a}{|x|^2}, $$ where $\lambda=\pm1$, $\alpha,b>0$ and $a>-\frac{(N-2)^2}{4}$. We first establish sufficient conditions for global existence and blow-up in $H^1_a(\mathbb{R}^N)$ for $\lambda=1$, using a Gagliardo-Nirenberg-type estimate. In the sequel, we study local and global well-posedness in $H^1_a(\mathbb{R}^N)$ in the $H^1$-subcritical case, applying the standard Strichartz estimates combined with the fixed point argument. The key to do that is to establish good estimates on the nonlinearity. Making use of these estimates, we also show a scattering criterion and construct a wave operator in $H^1_a(\mathbb{R}^N)$, for the mass-supercritical and energy-subcritical case.