On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping

Abstract

We consider the Cauchy problem for the $L^2$-critical nonlinear Schrödinger 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 $\Vert \nabla u(t){\Vert }_{L^2}$.