Some remarks on the inhomogeneous biharmonic NLS equation

TL;DR

This paper improves global well-posedness results for the inhomogeneous biharmonic nonlinear Schrödinger equation in certain dimensions and explores critical cases using Strichartz estimates and Hardy-Littlewood inequalities.

Contribution

It extends previous results by establishing improved global well-posedness and stability for the inhomogeneous biharmonic NLS in subcritical and critical cases.

Findings

Enhanced global well-posedness in dimensions 5, 6, 7.

Results on well-posedness and stability in the energy-critical case.

Application of Strichartz estimates and Hardy-Littlewood inequality.

Abstract

We consider the inhomogeneous biharmonic nonlinear Schr\"odinger equation $$ i u_t +\Delta^2 u+\lambda|x|^{-b}|u|^\alpha u = 0, $$ where $\lambda=\pm 1$ and $\alpha$, $b>0$. In the subctritical case, we improve the global well-posedness result obtained in \cite{GUZPAS} for dimensions $N=5,6,7$ in the Sobolev space $H^2(\mathbb{R}^N)$. The fundamental tools to establish our results are the standard Strichartz estimates related to the linear problem and the Hardy-Littlewood inequality. Results concerning the energy-critical case, that is, $\alpha=\frac{8-2b}{N-4}$ are also reported. More precisely, we show well-posedness and a stability result with initial data in the critical space $\dot{H}^2$.