On global well-posedness, scattering and other properties for infinity energy solutions to inhomogeneous NLS Equation

TL;DR

This paper establishes the global well-posedness and scattering for inhomogeneous nonlinear Schrödinger equations with energy solutions in Lorentz spaces, addressing cases where the potential is not in traditional Lebesgue spaces.

Contribution

It introduces a novel approach using Lorentz space estimates to prove global well-posedness and scattering for INLS with inhomogeneous potentials, extending previous results.

Findings

Proved global well-posedness in Lorentz spaces for certain parameters.

Established existence of self-similar solutions and wave operators.

Demonstrated asymptotic stability of solutions.

Abstract

In this work, we consider the inhomogeneous nonlinear Schr\"odinger (INLS) equation in $\mathbb{R}^n$ \begin{align} i\partial_t u + \Delta u + \gamma |x|^{-b}|u|^{\alpha} u = 0, \end{align} where $\gamma=\pm 1$, and $\alpha$ and $b$ are positive numbers. Our main focus is to estabilish the global well-posedness of the INLS equation in Lorentz spaces for $0<b<2$ and $\alpha<\frac{4-2b}{N-2}$. To achieve this, we use Strichartz estimates in Lorentz spaces $L^{r,q}(\R^n)$ combined with a fixed point argument. Working on Lorentz space setting instead the classical $L^p$ is motivated by the fact that the potential $|x|^{-b}$ does not belong the usual $L^p$-space. As a consequence of the ideas developed here on the global solution study we obtain some other properties for INLS, such as, existence of self-similar solutions, scattering, wave operators and assymptotic stability.