Periodic motions for multi-wells potentials and layers dynamic for the vector Allen-Cahn equation

Abstract

We consider a nonnegative potential W that vanishes on a finite set and study the existence of periodic orbits of the equationthat have the property of visiting neighborhoods of zeros of W in a given finite sequence. We give conditions for the existence of such orbits. After introducing the new variable x = ǫt, ǫ > 0 small, these orbits correspond to stationary solutions of the parabolic equationwith periodic boundary conditions.In the second paper of the paper we study solutions of this equation that, as the stationary solutions, have a layered structure.We derive a system of ODE that describes the dynamics of the layers and show that their motion is extremely slow.