On the convergence of a fourth order evolution equation to the Allen-Cahn equation

We construct special sequences of solutions to a fourth order nonlinear parabolic equation of Cahn-Hilliard/Allen-Cahn type, converging to the second order Allen-Cahn equation. We consider the evolution equation without boundary, as well as the stationary case on domains with Dirichlet boundary conditions. The proofs exploit the equivalence of the fourth order equation with a system of two second order elliptic equations with “good signs”.