Cahn–Hilliard equation with two spatial variables. Pattern formation

We consider the Cahn–Hilliard equation in the case where its solution depends on two spatial variables, with homogeneous Dirichlet and Neumann boundary conditions, and also periodic boundary conditions. For these three boundary value problems, we study the problem of local bifurcations arising when changing stability by spatially homogeneous equilibrium states. We show that the nature of bifurcations that lead to spatially inhomogeneous solutions is strongly related to the choice of boundary conditions. In the case of homogeneous Dirichlet boundary conditions, spatially inhomogeneous equilibrium states occur in a neighborhood of a homogeneous equilibrium state, depending on both spatial variables. An alternative scenario is realized in analyzing the Neumann problem and the periodic boundary value problem. In these, as a result of bifurcations, invariant manifolds formed by spatially inhomogeneous solutions occur. The dimension of these manifolds ranges from 1 to 3. In analyzing three boundary value problems, we use methods of infinite-dimensional dynamical system theory and asymptotic methods. Using the integral manifold method together with the techniques of normal form theory allows us to analyze the stability of bifurcating invariant manifolds and also to derive asymptotic formulas for spatially inhomogeneous solutions forming these manifolds.