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

where $\lambda \in \mathbb{C}$, $d\ge 3$, $1\le s<\frac{d}{2}$, $0<b<\min \{4,2+\frac{d}{2}-s\}$ 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 by An–Kim–Ryu [Discrete Contin. Dyn. Syst. Ser. B 29 (2024), 3326–3345]. This result also extends the ones of Saanouni–Ghanmi [Adv. Oper. Theory 9 (2024), article no. 1] and Saanouni–Peng [Mediterr. J. Math. 20 (2023), article no. 170] by extending the validity of $d$, $b$, and $s$. Finally, we give the well-posedness result in the homogeneous Sobolev spaces ${\dot{H}}^s$.