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| We consider the nonlinear Schr\"{o}dinger equations with inverse-square potentials $a|x|^{-2}$: \[i\frac{\partial u}{\partial t}=-\Delta{u}+\frac{a}{|x|^{2}}u+f(u)\;\text{in}\quad\mathbb{R}\times\mathbb{R}^{N}\] where $i=\sqrt{-1}$, $N\ge 3$ and $a\ge-(N-2)^{2}/4$. The feature is the presence of a strongly singular potential $a|x|^{-2}$; note that $-\Delta$ and $a|x|^{-2}$ are the same scale symmetry and hence scaling argument can not be applied to $P_{a}:=-\Delta+a|x|^{-2}$. The restriction on $a$ follows from the selfadjointness of $P_{a}$ in the sense of form-sum in $L^{2}(\mathbb{R}^{N})$. In this talk we show the global existence and blow-up in finite time. |
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