Blowup for the nonlinear Schr\"odinger equation with an inhomogeneous damping term in the $L^2$ critical case

We consider the nonlinear Schr"odinger equation with $L^2$-critical exponent and an inhomogeneous damping term. By using the tools developed by Merle and Raphael, we prove the existence of blowup phenomena in the energy space $H^1(\mathbb{R})$.