A note on the $H^s$-critical inhomogeneous nonlinear Schrödinger equation

Abstract

In this paper, we consider the Cauchy problem for the $H^s$-critical inhomogeneous nonlinear Schrödinger (INLS) equation

where $n\in \mathbb{N}$, $0\le s<\frac{n}{2}$, $0<b<\min \{2,n-s,1+\frac{n-2s}{2}\}$ and $f(u)$ is a nonlinear function that behaves like $\lambda |u{|}^{\sigma }u$ with $\lambda \in \mathbb{C}$ and $\sigma =\frac{4-2b}{n-2s}$. First, we establish the local well-posedness as well as the small data global well-posedness in $H^s({\mathbb{R}}^n)$ for the $H^s$-critical INLS equation by using the contraction mapping principle based on the Strichartz estimates in Sobolev–Lorentz spaces. Next, we obtain some standard continuous dependence results for the $H^s$-critical INLS equation. Our results about the well-posedness and standard continuous dependence for the $H^s$-critical INLS equation improve the ones of Aloui–Tayachi [Discrete Contin. Dyn. Syst. 41 (2021), 5409–5437] and An–Kim [Evol. Equ. Control Theory 12 (2023), 1039–1055] by extending the validity of $s$ and $b$. Based on the local well-posedness in $H^1({\mathbb{R}}^n)$, we finally establish the blow-up criteria for $H^1$-solutions to the focusing energy-critical INLS equation. In particular, we prove the finite time blow-up for finite-variance, radially symmetric or cylindrically symmetric initial data.