Global well-posedness and blow-up on the energy space for the Inhomogeneous Nonlinear Schr\"odinger Equation

TL;DR

This paper investigates the supercritical inhomogeneous nonlinear Schrödinger equation, establishing conditions for global solutions and blow-up in the energy space using a Gagliardo-Nirenberg type estimate.

Contribution

It introduces a new Gagliardo-Nirenberg estimate and applies it to determine criteria for global existence and blow-up in the INLS equation.

Findings

Derived a Gagliardo-Nirenberg type inequality for INLS.

Established sufficient conditions for global solutions.

Identified criteria for finite-time blow-up.

Abstract

We consider the supercritical inhomogeneous nonlinear Schr\"odinger 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 $H^1(\mathbb{R}^N)$.