Sobolev-Lorentz spaces with an application to the inhomogeneous biharmonic NLS equation

Abstract

We consider the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation \[iu_{t} +Δ^{2} u=λ|x|^{-b}|u|^σu,\;u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $\lambda \in \mathbb{R}$, $d\in \mathbb{N}$, $0\le s<\min \left\{2+\frac{d}{2},d\right\}$, $0<b<\min \left\{4,d-s,2+\frac{d}{2}-s\right\}$ and $0<\sigma \le {\sigma }_c(s)$ with $\sigma <\infty$. Here ${\sigma }_c(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and ${\sigma }_c(s)=\infty$ if $s\ge \frac{d}{2}$. First, we give some remarks on Sobolev-Lorentz spaces and extend the chain rule under Lorentz norms for the fractional Laplacian $(-\Delta {)}^{s/2}$ with $s\in (0,1]$ established by [Discrete Contin. Dyn. Syst. 41 (2021) 5409-5437] to any $s>0$. Applying this estimate and the contraction mapping principle based on Strichartz estimates in Lorentz spaces, we then establish the local well-posedness in $H^s$ for the IBNLS equation in both of subcritical case $\sigma <{\sigma }_c(s)$ and critical case $\sigma ={\sigma }_c(s)$. We also prove that the IBNLS equation is globally well-posed in $H^s$, if the initial data is sufficiently small and $\frac{8-2b}{d}\le \sigma \le {\sigma }_c(s)$ with $\sigma <\infty$.

22 pages. arXiv admin note: text overlap with arXiv:2206.06690