Global well-posedness and critical norm concentration for inhomogeneous biharmonic NLS

TL;DR

This paper investigates the well-posedness, global existence, and norm concentration phenomena for the inhomogeneous biharmonic nonlinear Schrödinger equation in higher dimensions, providing new inequalities and embedding results.

Contribution

It establishes local and global well-posedness results, introduces a Gagliardo-Nirenberg inequality for this equation, and analyzes norm concentration during finite-time blow-up.

Findings

Well-posedness in $ ext{dot} H^{s_c} igcap ext{dot} H^2$ for $N \\geq 5$

Conditions for global existence based on a Gagliardo-Nirenberg inequality

Norm concentration phenomena during finite-time blow-up

Abstract

We consider the inhomogeneous biharmonic nonlinear Schr\"odinger (IBNLS) equation in $\mathbb{R}^N$, $$i \partial_t u +\Delta^2 u -|x|^{-b} |u|^{2\sigma}u = 0,$$ where $\sigma>0$ and $b>0$. We first study the local well-posedness in $\dot H^{s_c}\cap \dot H^2 $, for $N\geq 5$ and $0<s_c<2$, where $s_c=\frac{N}{2}-\frac{4-b}{2\sigma}$. Next, we established a Gagliardo-Nirenberg type inequality in order to obtain sufficient conditions for global existence of solutions in $\dot H^{s_c}\cap \dot H^2$ with $0\leq s_c<2$. Finally, we study the phenomenon of $L^{\sigma_c}$-norm concentration for finite time blow up solutions with bounded $\dot H^{s_c}$-norm, where $\sigma_c=\frac{2N\sigma}{4-b}$. Our main tool is the compact embedding of $\dot L^p\cap \dot H^2$ into a weighted $L^{2\sigma+2}$ space, which may be seen of independent interest.