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

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