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We consider the Cauchy problem for the $L^{2}$-critical nonlinearSchr\"{o}dinger equation with a nonlinear damping. According to the power ofthe damping term, we prove the global existence or the existence of finite timeblowup dynamics with the log-log blow-up speed for $\|\nabla u(t)\|_{L^2}$.

On the $L^{2}$-critical nonlinear Schr\"odinger Equation with a nonlinear damping