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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 aGagliardo-Nirenberg type estimate and use it to establish sufficient conditionsfor global existence and blow-up in $H^1(\mathbb{R}^N)$.

Global well-posedness and blow-up on the energy space for the Inhomogeneous Nonlinear Schr\"odinger Equation