Small data global well-posedness for the inhomogeneous biharmonic NLS in Sobolev spaces

In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS) $iu_{t} +Δ^{2} u=λ|x|^{-b}|u|^σu,u(0)=u_{0} \in H^{s} (\mathbb R^{d}),$ where $λ\in \mathbb R$, $d\in \mathbb N$, $0<s<\min \{2+\frac{d}{2},\frac{3}{2}d\}$ and $0<b<\min\{4,d,\frac{3}{2}d-s,\frac{d}{2}+2-s\}$. Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is globally well-posed in $H^{s}(\mathbb R^{d})$ if $\frac{8-2b}{d}<σ< σ_{c}(s)$ and the initial data is sufficiently small, where $σ_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and $σ_{c}(s)=\infty$ if $s\ge \frac{d}{2}$.

No reviews yet.

No comments yet.