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

Abstract

In this work, we consider the inhomogeneous nonlinear Schrödinger (INLS) equation in $R^n$$i{\partial }_{t}u+\Delta u+\gamma \left|x{|}^{-b}\right|u{|}^{\alpha }u=0,$ where $\gamma =\pm 1$, and α and b are positive numbers. Our main focus is to establish 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 $L^{r,q}\left(R^n\right)$ 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 asymptotic stability.