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

Abstract

In this paper, we investigate the Cauchy problem for the $H^s$-critical inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation \[iu_{t}\pm Δ^{2} u=λ|x|^{-b}|u|^σu,~u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $\hat{I}»\in \mathbb{C}$, $d\ge 3$, $1\le s<\frac{d}{2}$, $0<b<\min \left\{4,2+\frac{d}{2}-s\right\}$ and $\ddot{I}ƒ=\frac{8-2b}{d-2s}$. 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 ${\dot{H}}^s$.