Finite-time blowup for a complex Ginzburg-Landau equation

We prove that negative energy solutions of the complex Ginzburg-Landau equation $e^{-i\theta }{u}_t=\mathrm{\Delta }u+\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u$ blow up in finite time, where \alpha >0 and \pi /2<\theta <\pi /2. For a fixed initial value $u(0)$, we obtain estimates of the blow-up time $T_{max}^{\theta }$ as $\theta \to \pm \pi /2 $\mathrm{.}Itturnsoutthat$T_{max}^\theta $ stays bounded (respectively, goes to infinity) as $\theta \to \pm \pi \mathrm{/}2$ in the case where the solution of the limiting nonlinear Schr"odinger equation blows up in finite time (respectively, is global).