Blow-up of non-radial solutions for the $L^2$ critical inhomogeneous NLS equation

Abstract

We consider the $L^2$ critical inhomogeneous nonlinear Schrödinger (INLS) equation in ${\mathbb{R}}^N$ $$i{\partial }_{t}u+\Delta u+|x{|}^{-b}|u{|}^{\frac{4-2b}{N}}u=0,$$ where $N\ge 1$ and $0<b<2$. We prove that if $u_0\in H^1({\mathbb{R}}^N)$ satisfies $E[u_0]<0$, then the corresponding solution blows-up in finite time. This is in sharp contrast to the classical $L^2$ critical NLS equation where this type of result is only known in the radial case for $N\ge 2$.

9 pages

References in corpus (3)

• Blow-up of radial solutions for the intercritical inhomogeneous NLS equation
• Finite time/Infinite time blow-up behaviors for the inhomogeneous nonlinear Schrödinger equation
• Blow-up solutions of the intercritical inhomogeneous NLS equation: the non-radial case