We consider the $L^2$ critical inhomogeneous nonlinear Schr\"odinger (INLS) equation in $\mathbb{ R}^N$ $$ i \partial_t u +\Delta u +|x|^{ -b} |u|^{ \frac{ 4-2b}{ N}}u = 0, $$ where $N\geq 1$ and $0<b<2$. We prove that if $u_0\in H^1(\mathbb{ R}^N)$ satisfies $E[u_0]<0$, then the corresponding solution blows-up in finite time. This is in sharp contrast to the classical $L^2$ critical NLS equation where this type of result is only known in the radial case for $N\geq 2$.
Categories: math.AP