Approximation and attractivity properties of the degenerated Ginzburg–Landau equation

We are interested in spatially extended pattern forming systems close to the threshold of the first instability in case when the so-called degenerated Ginzburg–Landau equation takes the role of the classical Ginzburg–Landau equation as the amplitude equation of the system. This is the case when the relevant nonlinear terms vanish at the bifurcation point. Here we prove that in this situation every small solution of the pattern forming system develops in such a way that after a certain time it can be approximated by the solutions of the degenerated Ginzburg–Landau equation. In this paper we restrict ourselves to a Swift–Hohenberg–Kuramoto–Shivashinsky equation as a model for such a pattern forming system.