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 $\|\nabla u(t)\|_{L^2}$. 
Mohamad Darwich (LMPT)
math.AP