Global well-posedness and blow-up on the energy space for the Inhomogeneous Nonlinear Schrödinger Equation

Abstract

We consider the supercritical inhomogeneous nonlinear Schrödinger equation (INLS) $$i{\partial }_{t}u+\Delta u+|x{|}^{-b}|u{|}^{2\sigma }u=0,$$ where $(2-b)/N<\sigma <(2-b)/(N-2)$ and $0<b<\min \{2,N\}$. We prove a Gagliardo-Nirenberg type estimate and use it to establish sufficient conditions for global existence and blow-up in $H^1({\mathbb{R}}^N)$.

arXiv comment14 pages