Local Dynamics of Cahn–Hilliard Equation

Abstract

For the generalized Cahn–Hilliard equation it is shown that in some domain of the phase field its local dynamics is described with the help of Andronov–Hopf bifurcation. The appropriate normal form is given which defines the behavior of the solutions in this domain of the phase field. The problem is considered with the large coefficient of advection which leads to the infinite-dimensional critical case in the problem about balance state stability. It is shown that the local dynamics of the initial boundary-value problem is determined by the nonlocal behavior of the solutions of specially constructed simpler nonlinear boundary-value problem.