Motion of interfaces for hyperbolic variations of the Allen-Cahn equation

The Allen-Cahn equation is a (parabolic) reaction-diffusion equation with a balanced bistable reaction term, which describes phase transition processes. It is well-known that when the diffusion coefficient is very small, the solutions exhibit very interesting phenomena. In the one-dimensional case, we have an example of metastable dynamics, while in the multi-dimensional case the Allen-Cahn equation is strictly related to the mean curvature flow. In this paper we discuss such phenomena in the case of some hyperbolic variations of the Allen-Cahn equation. In particular, in the one-dimensional case we focus the attention on the assumptions needed to have metastability and we show some numerical solutions in the case such assumptions are not satisfied.