Some stability results for the complex Ginzburg-Landau equation

Using some classical methods of dynamical systems, stability results and asymptotic decay of strong solutions for the complex Ginzburg-Landau equation (CGL), $$ \partial_t u = (a + i\alpha) \Delta u - (b + i \beta) |u|^\sigma u + k u, \,\, t > 0,\,\, x\in \Omega, $$ with $a>0, \alpha, b, \beta, k \in \mathbb{R}$, are obtained. Moreover, we show the existence of bound-states under certain conditions on the parameters and on the domain. We conclude with the proof of asymptotic stability of these bound-states when $\Omega=\mathbb{R}$ and $-k$ large enough.