AUTHOR=Medina David , Padilla Pablo 

TITLE=The Global Attractor of the Allen-Cahn Equation on the Sphere

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=Volume 6 - 2020

YEAR=2020

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2020.00020

DOI=10.3389/fams.2020.00020

ISSN=2297-4687

ABSTRACT=In this paper we study the attractor of a parabolic semiflow generated by a PDE with a nonlinear term given by a bistable potential, in an oval surface; the Allen-Cahn equation being a prototypical example. An additional constraint arising from geometric considerations will be introduced. The exis- tence of a global attractor will be obtained by modifying standard techniques in order to handle the constraint. We provide numerical simulations using Galerkin method and conjecture that the transition layer 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.