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

Abstract

In this work, we consider the inhomogeneous nonlinear Schrödinger (INLS) equation in ${\mathbb{R}}^n$ \begin{align} i\partial_t u + Δu + γ|x|^{-b}|u|^α 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 $L^{r,q}({\mathbb{R}}^n)$ 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.