The Global Attractor of the Allen-Cahn Equation on the Sphere - Medina, David; Padilla, Pablo (Frontiers in Applied Mathematics and Statistics, 2020)
In this paper we study the attractor of a parabolic semiflow generated by a singularly perturbed PDE with a non-linear term given by a bistable potential, in an oval surface; the Allen-Cahn equation being a prototypical example. An additional constraint motivated by a variational principle for closed geodesics originally proposed by Poincaré arising from geometric considerations is introduced. The existence of a global attractor is established by modifying standard techniques in order to handle the constraint. Based on previous work on the elliptic case, it is known that when the perturbation parameter tends to zero, minimal energy solutions exhibit a sharp interface concentrated on a closed geodesic. We provide numerical simulations using Galerkin's method. Based on the analytical and numerical results we conjecture that, when the perturbation parameter tends to zero and for large times, the transition layers of the solutions of this PDE consists of closed geodesics or a union of arcs of such geodesics, thus characterizing the structure of the attractor.