Global well-posedness and critical norm concentration for inhomogeneous biharmonic NLS

Abstract

We consider the inhomogeneous biharmonic nonlinear Schr dinger (IBNLS) equation in ${\mathbb {R}}^N$,

$$\begin{aligned} i \partial _t u +\Delta ^2 u -|x|^{-b} |u|^{2\sigma }u = 0, \end{aligned}$$

where $\sigma > 0$ and $b > 0$. We first study the local well-posedness in ${\dot{H}}^{s_c}\cap \dot{H}^2$, for $N\ge 5$ and $0<s_c<2$, where $s_c=\frac{N}{2}-\frac{4-b}{2\sigma }$. Next, we established a Gagliardo-Nirenberg type inequality in order to obtain sufficient conditions for global existence of solutions in $\dot{H}^{s_c}\cap \dot{H}^2$ with $0\le s_c<2$. Finally, we study the phenomenon of $L^{\sigma _c}$-norm concentration for finite time blow up solutions with bounded $\dot{H}^{s_c}$-norm, where $\sigma _c=\frac{2N\sigma }{4-b}$. Our main tool is the compact embedding of $\dot{L}^p\cap \dot{H}^2$ into a weighted $L^{2\sigma +2}$ space, which may be seen of independent interest.