Joint BCAM-UPV/EHU Analysis and PDE seminar: Heteroclinic solutions of the Allen-Cahn equation and mean curvature flows

The Allen-Cahn equation is a semilinear partial differential equation which models phase transition and separation phenomena and which provides a regularization for the mean curvature flow (MCF), one of the most studied geometric flows. 

In this talk, we combine analytic, geometric and topological strategies to obtain existence results for eternal solutions of this parabolic PDE connecting unstable nontrivial stationary solutions, namely heteroclinic solutions, in certain compact manifolds. In the concrete setting of a 3-dimensional round sphere, we describe the space of all low energy eternal solutions and explain how they can be used to construct geometrically interesting MCFs. 

This is joint work with Jingwen Chen (University of Pennsylvania).