  We consider the inhomogeneous biharmonic nonlinear Schr\"odinger (IBNLS)
equation in $\mathbb{R}^N$, $$i \partial_t u +\Delta^2 u -|x|^{-b}
|u|^{2\sigma}u = 0,$$ where $\sigma>0$ and $b>0$. We first study the local
well-posedness in $\dot H^{s_c}\cap \dot H^2 $, for $N\geq 5$ 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 $0\leq
s_c<2$. 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.
