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Dynamical bifurcation for the Kuramoto-Sivashinsky equation. (English) Zbl 1209.35017
This study concerns the one-dimensional Kuramoto-Sivashinsky equation with periodic boundary condition, and with the constraint that the mean over a period is 0. The aim is to study the bifurcation from the trivial solution corresponding to eigenvalues of the linearized equation. Using the attractor bifurcation theory of Tian Ma and S. Wang [Chin. Ann. Math., Ser. B 26, No. 2, 185–206 (2005; Zbl 1193.37105)], T. Ma and S. Wang [Commun. Pure Appl. Anal. 2, No. 4, 591–599 (2003; Zbl 1210.37056)], bifurcation of a nontrivial attractor from the trivial attractor 0 is proved when the eigenvalue $\lambda =1$ is crossed, and a characterization of its homotopy type is given, both in the general case and when the problem is considered in the space of odd functions.
##### MSC:
 35B32 Bifurcation (PDE)