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), $${\mathrm{\partial }}_{t}u=(a+i\alpha )\mathrm{\Delta }u-(b+i\beta )\mathrm{\mid }u{\mathrm{\mid }}^{\sigma }u+ku,t>0,x\in \mathrm{\Omega },$$ 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 $\mathrm{\Omega }=\mathbb{R}$ and $-k$ large enough.