Motion of interfaces for a damped hyperbolic Allen-Cahn equation

Abstract

Consider the Allen-Cahn equation $u_t={\epsilon }^2\Delta u-F^{\prime }(u)$, where $F$ is a double well potential with wells of equal depth, located at $\pm 1$. There are a lot of papers devoted to the study of the limiting behavior of the solutions as the diffusion coefficient $\epsilon \to 0^{+}$, 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 $\epsilon \to 0^{+}$ also in the hyperbolic framework.

46 pages, 6 figures

References in corpus (3)

• Slow motion for a hyperbolic variation of Allen-Cahn equation in one space dimension
• The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system
• Slow motion for one-dimensional nonlinear damped hyperbolic Allen-Cahn systems