论文标题
Sobolev空间中不均匀的Biharmonic非线性schrödinger方程的本地适应性
Local well-posedness for the inhomogeneous biharmonic nonlinear Schrödinger equation in Sobolev spaces
论文作者
论文摘要
在本文中，我们研究了不均匀的biharmonic norearearearearearearearearearearear（ibnls）方程的库奇问题\ [iu_ {t} +δ^{2} u =λ| x | x | x | x | x |^{ - b} | r^{d}），\]其中$ d \ in \ mathbb n $，$ s \ ge 0 $，$ 0 <b <4 $，$σ> 0 $和$λ\ in \ mathbb r $。在非线性术语的某些规律性假设下，我们证明IBNLS方程在$ h^{s}（\ Mathbb r^{d}）$ in \ mathbb n $，$ 0 \ {2+ \ frac {d} {2}，\ frac {3} {2} {2} d \} $，$ 0 <b <b <\ min \ {4，d，d，\ frac {3} {2} {2} d-s，d-s，\ frac {d} d}+2}+2-s+2-s+2-s+2-s \} $ 0 <这里$σ_{c}（s）= \ frac {8-2b} {d-2s} $如果$ s <\ frac {d} {2} {2} $和$σ_{c}（c}（s）= \ infty $ s = \ infty $如果$ s \ ge \ ge \ ge \ frac {d} {d} {2} {2} {2} $。我们的本地适合性结果改善了Guzmán-Pastor的结果[非线性肛门。现实世界应用。 56（2020）103174]和Liu-Zhang [J.微分方程296（2021）335-368]通过扩展$ s $和$ b $的有效性。
In this paper, we study 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 $d\in \mathbb N$, $s\ge 0$, $0<b<4$, $σ>0$ and $λ\in \mathbb R$. 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 $0<σ< σ_{c}(s)$. Here $σ_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and $σ_{c}(s)=\infty$ if $s\ge \frac{d}{2}$. Our local well-posedness result improves the ones of Guzmán-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 $b$.
