  In this paper, we consider the complex Ginzburg--Landau equation $u_t =
e^{i\theta} [\Delta u + |u|^\alpha u] + \gamma u$ on ${\mathbb R}^N $, where
$\alpha >0$, $\gamma \in \R$ and $-\pi /2<\theta <\pi /2$. By convexity
arguments we prove that, under certain conditions on $\alpha ,\theta ,\gamma $,
a class of solutions with negative initial energy blows up in finite time.
