Regular Strichartz estimates in Lorentz-type spaces with application to the $H^s$-critical inhomogeneous biharmonic NLS equation

TL;DR

This paper develops regular Strichartz estimates in Lorentz-type spaces and applies them to prove well-posedness and scattering for the $H^s$-critical inhomogeneous biharmonic NLS equation, extending previous results to less regular nonlinearities.

Contribution

It introduces Lorentz-type space estimates for the linear equation and uses these to establish well-posedness and scattering results for the critical IBNLS with weaker regularity assumptions.

Findings

Established local and global well-posedness in $H^s$

Proved scattering for small initial data

Extended validity of results to broader ranges of parameters

Abstract

In this paper, we investigate the Cauchy problem for the $H^s$-critical inhomogeneous biharmonic nonlinear Schr\"{o}dinger (IBNLS) equation \[iu_{t}\pm \Delta^{2} u=\lambda |x|^{-b}|u|^{\sigma}u,~u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $\lambda\in \mathbb C$, $d\ge 3$, $1\le s<\frac{d}{2}$, $0<b<\min \left\{4,2+\frac{d}{2}-s \right\}$ and $\sigma=\frac{8-2b}{d-2s}$. First, we study the properties of Lorentz-type spaces such as Besov-Lorentz spaces and Triebel-Lizorkin-Lorentz spaces. We then derive the regular Strichartz estimates for the corresponding linear equation in Lorentz-type spaces. Using these estimates, we establish the local well-posedness as well as the small data global well-posedness and scattering in $H^s$ for the $H^s$-critical IBNLS equation under less regularity assumption on the nonlinear term than in the recent work \cite{AKR24}. This result also extends the ones of \cite{SP23,SG24} by extending the validity of $d$, $b$ and $s$. Finally, we give the well-posedness result in the homogeneous Sobolev spaces $\dot{H}^s$.