Dynamical bifurcation for the Kuramoto-Sivashinsky equation (Q622385)

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 \textit{Tian Ma} and \textit{S. Wang} [Chin. Ann. Math., Ser. B 26, No. 2, 185--206 (2005; Zbl 1193.37105)], \textit{T. Ma} and \textit{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.