Motion of interfaces for a damped hyperbolic Allen-Cahn equation

Abstract

Consider the Allen-Cahn equation $u_t=\varepsilon^2\Delta u-F'(u)$, where $F$ is a double well potential with wells of equal depth, located at $\pm1$. There are a lot of papers devoted to the study of the limiting behavior of the solutions as the diffusion coefficient $\varepsilon\to0^+$, and it is well known that, if the initial datum $u(\cdot,0)$ takes the values $+1$ and $-1$ in the regions $\Omega_+$ and $\Omega_-$, then the "interface" connecting $\Omega_+$ and $\Omega_-$ moves with normal velocity equal to the sum of its principal curvatures, i.e. the interface moves by mean curvature flow. This paper concerns with the motion of the inteface for a damped hyperbolic Allen-Cahn equation, in a bounded domain of $\mathbb{R}^n$, for $n=2$ or $n=3$. In particular, we focus the attention on radially simmetric solutions, studying in detail the differences with the classic parabolic case, and we prove that, under appropriate assumptions on the initial data $u(\cdot,0)$ and $u_t(\cdot,0)$, the interface moves by mean curvature as $\varepsilon\to0^+$ also in the hyperbolic framework.