On the Nature of Local Bifurcations of the Kuramoto–Sivashinsky Equation in Various Domains

We consider a nonlinear parabolic partial differential equation in the case where the unknown function depends on two spatial variables and time, which is a generalization of the well-known Kuramoto–Sivashinsky equation. We consider homogeneous Dirichlet boundary-value problems for this equation. We examine local bifurcations when spatially homogeneous equilibrium states change stability. We show that post-critical bifurcations are realized in the boundary-value problems considered. We obtain asymptotic formulas for solutions and examine the stability of spatially inhomogeneous solutions.