Local well-posedness for the inhomogeneous biharmonic nonlinear Schr\"{o}dinger equation in Sobolev spaces

TL;DR

This paper establishes local well-posedness for the inhomogeneous biharmonic nonlinear Schrödinger equation in Sobolev spaces, extending previous results by broadening the parameter ranges for regularity and inhomogeneity.

Contribution

It provides new well-posedness results for the IBNLS equation, extending the parameter ranges for Sobolev regularity and inhomogeneity compared to prior work.

Findings

Proves local well-posedness in H^s for specific s, b, and σ ranges.

Extends previous results by relaxing regularity and inhomogeneity constraints.

Improves the understanding of solution behavior for the IBNLS equation.

Abstract

In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schr\"{o}dinger (IBNLS) equation \[iu_{t} +\Delta^{2} u=\lambda |x|^{-b}|u|^{\sigma}u,~u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $d\in \mathbb N$, $s\ge 0$, $0<b<4$, $\sigma>0$ and $\lambda \in \mathbb R$. Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is locally well-posed in $H^{s}(\mathbb R^{d})$ if $d\in \mathbb N$, $0\le s <\min \{2+\frac{d}{2},\frac{3}{2}d\}$, $0<b<\min\{4,d,\frac{3}{2}d-s,\frac{d}{2}+2-s\}$ and $0<\sigma< \sigma_{c}(s)$. Here $\sigma_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and $\sigma_{c}(s)=\infty$ if $s\ge \frac{d}{2}$. Our local well-posedness result improves the ones of Guzm\'{a}n-Pastor [Nonlinear Anal. Real World Appl. 56 (2020) 103174] and Liu-Zhang [J. Differential Equations 296 (2021) 335-368] by extending the validity of $s$ and $b$.