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

We consider the inhomogeneous biharmonic nonlinear Schr dinger (IBNLS) equation in $${\mathbb {R}}^N$$ , $$\begin{aligned} i \partial _t u +\Delta ^2 -|x|^{-b} |u|^{2\sigma }u = 0, \end{aligned}$$ where $$\sigma > 0$$ and $$b . first study local well-posedness $${\dot{H}}^{s_c}\cap \dot{H}^2 $$ for $$N\ge 5$$ $$0<s_c<2$$ $$s_c=\frac{N}{2}-\frac{4-b}{2\sigma }$$ Next, we established a Gagliardo-Nirenberg type inequality order to obtain sufficient conditions global existence of solutions $$\dot{H}^{s_c}\cap \dot{H}^2$$ with $$0\le s_c<2$$ Finally, phenomenon $$L^{\sigma _c}$$ -norm concentration finite time blow up bounded $$\dot{H}^{s_c}$$ -norm, _c=\frac{2N\sigma }{4-b}$$ Our main tool is compact embedding $$\dot{L}^p\cap into weighted $$L^{2\sigma +2}$$ space, which may be seen independent interest.