Finite-time blowup for a complex Ginzburg-Landau equation

We prove that negative energy solutions of the complex Ginzburg-Landau equation $e^{-iθ} u_t = Δu+ |u|^α u$ blow up in finite time, where α>0 and π/2<θ<π/2. For a fixed initial value $u(0)$, we obtain estimates of the blow-up time $T_{max}^θ$ as $θ\to \pm π/2$. It turns out that $T_{max}^θ$ stays bounded (respectively, goes to infinity) as $θ\to \pm π/2$ in the case where the solution of the limiting nonlinear Schrödinger equation blows up in finite time (respectively, is global).

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