Existence, stability and the number of two-dimensional invariant manifolds for the convective Cahn–Hilliard equation

Abstract

We study the well-known generalised version of the nonlinear Cahn–Hilliard evolution equation, supplemented with periodic boundary conditions. We study local bifurcations in the vicinity of spatially homogeneous equilibrium states. We show the possibility of the existence of a finite or countable set of equilibrium states of the boundary value problem under study, in the vicinity of which, if appropriate conditions are met, there exist two-dimensional invariant manifolds filled with solutions that are periodic in the evolutionary variable. Moreover, we derive asymptotic formulas for these periodic solutions. Finally, we study the stability of invariant manifolds and the solutions belonging to them.

In order to analyse the bifurcation problem, we used methods from the theory of dynamical systems with infinite-dimensional phase, namely the method of invariant manifolds and the method of normal forms.