In this paper, we consider some hyperbolic variants of the mass conserving Allen-Cahn equation, which is a nonlocal reaction-diffusion equation, introduced (as a simpler alternative to the Cahn-Hilliard equation) to describe phase separation in binary mixtures. In particular, we focus our attention on the metastable dynamics of some solutions to the equation in a bounded interval of the real line with homogeneous Neumann boundary conditions. It is shown that the evolution of profiles with $N+1$ transition layers is very slow and we derive a system of ODEs, which describes the exponentially slow motion of the layers. A comparison with the classical Allen-Cahn and Cahn-Hilliard equations and theirs hyperbolic variations is also performed.

中文翻译：

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一维空间中质量守恒 Allen-Cahn 方程的双曲变体的亚稳态动力学

在本文中，我们考虑了质量守恒 Allen-Cahn 方程的一些双曲线变体，这是一个非局部反应扩散方程，引入（作为 Cahn-Hilliard 方程的更简单替代方案）来描述二元混合物中的相分离。特别地，我们将注意力集中在具有齐次 Neumann 边界条件的实线的有界区间内方程的一些解的亚稳态动力学。结果表明，具有 $N+1$ 过渡层的剖面演化非常缓慢，我们推导出了一个 ODE 系统，它描述了层的指数慢运动。还进行了与经典 Allen-Cahn 和 Cahn-Hilliard 方程及其双曲线变化的比较。