On the inhomogeneous biharmonic nonlinear Schr\"odinger equation: local, global and stability results

TL;DR

This paper investigates the inhomogeneous biharmonic nonlinear Schrödinger equation, establishing local and global well-posedness and stability results in various Sobolev spaces using Strichartz estimates.

Contribution

It provides new well-posedness and stability results for the IBNLS in different regimes, extending understanding of its mathematical properties.

Findings

Established local and global well-posedness in $H^s$ for $s=0,2$ in the subcritical case.

Proved stability in $H^2$ for mass-supercritical and energy-subcritical regimes.

Utilized Strichartz estimates to derive key results.

Abstract

We consider the inhomogeneous biharmonic nonlinear Schr\"odinger equation (IBNLS) $$ i u_t +\Delta^2 u+\lambda|x|^{-b}|u|^\alpha u = 0, $$ where $\lambda=\pm 1$ and $\alpha$, $b>0$. We show local and global well-posedness in $H^s(\mathbb{R}^N)$ in the $H^s$-subcritical case, with $s=0,2$. Moreover, we prove a stability result in $H^2(\mathbb{R}^N)$, in the mass-supercritical and energy-subcritical case. The fundamental tools to prove these results are the standard Strichartz estimates related to the linear problem.