Finite-time blowup for a complex Ginzburg-Landau equation
Abstract
We prove that negative energy solutions of the complex Ginzburg--Landau equation $e^{-i\theta } u_t = \Delta u+ |u|^\alpha u$ 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_{\mathrm{max}}^\theta $ as $\theta \to \pm \pi /2 $. It turns out that $T_{\mathrm{max}}^\theta $ stays bounded (respectively, goes to infinity) as $\theta \to \pm \pi /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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