Well-posedness and scattering of odd solutions for the defocusing INLS in one dimension

Abstract: We consider the defocusing inhomogeneous nonlinear Schr\"{o}dinger equation $i\partial_tu+\Delta u= |x|^{-b}|u|^{\alpha}u,$ where $0<b<1$ and $0<\alpha<\infty$. This problem has been extensively studied for initial data in $H^1(\R^N)$ with $N\geq 2$. However, in the one-dimensional setting, due to the difficulty in dealing with the singularity factor $|x|^{-b}$, the well-posedness and scattering in $H^1(\R)$ are scarce, and almost known results have been established in $H^s(\R)$ with $s<1$. In this paper, we focus on the odd initial data in $H^1(\R)$. For this case, we establish local well-posedness for $0<\alpha<\infty$, as well as global well-posedness and scattering for $4-2b<\alpha<\infty$, which corresponds to the mass-supercritical case. The key ingredient is the application of the one-dimensional Hardy inequality for odd functions to overcome the singularity induced by $|x|^{-b}$. Our proof is based on the Strichartz estimates and employs the concentration-compactness/rigidity method developed by Kenig-Merle as well as the technique for handling initial data living far from the origin, as proposed by Miao-Murphy-Zheng. Our results fill a gap in the theory of well-posedness and energy scattering for the inhomogeneous nonlinear Schr\"{o}dinger equation in one dimension.