  We consider the supercritical inhomogeneous nonlinear Schr\"odinger equation
(INLS)
  $$i\partial_t u+\Delta u+|x|^{-b}|u|^{2\sigma}u=0,$$ where
$(2-b)/N<\sigma<(2-b)/(N-2)$ and $0<b<\min\{2,N\}$. We prove a
Gagliardo-Nirenberg type estimate and use it to establish sufficient conditions
for global existence and blow-up in $H^1(\mathbb{R}^N)$.
