SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2624-2654, January 2020.
In this paper, we study the generation of interface for the solution of the mass conserved Allen--Cahn equation involving a nonlocal integral term. We show that, for a rather general class of initial functions that are independent of $\varepsilon$, the solution generally develops a steep transition layer of thickness $O(\varepsilon^\gamma)$ $(0 <\gamma\le 1)$ at a certain time of order $\varepsilon^2 |\ln \varepsilon|$. In some cases, we prove that the thickness of the interface is exactly of order $\varepsilon$, which is the optimal thickness estimate. We note that the comparison principle does not hold for our equation because of the nonlocal term so that the methods that were employed in the earlier studies of the standard Allen--Cahn equation do not work. We will therefore take a different approach, which is based on the fine analysis of the long-time behavior of the corresponding nonlocal ODEs and some energy estimates.

中文翻译：

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守恒Allen-Cahn方程解的界面生成

SIAM数学分析期刊，第52卷，第3期，第2624-2654页，2020年1月。
在本文中，我们研究了涉及非局部积分项的质量守恒Allen-Cahn方程解的界面的生成。我们表明，对于独立于$ \ varepsilon $的相当通用的初始函数类，该解决方案通常会形成一个陡峭的过渡层，其厚度为$ O（\ varepsilon ^ \ gamma）$（0 <\ gamma \ le 1 ）$在订购$ \ varepsilon ^ 2 | \ ln \ varepsilon | $的某个时间。在某些情况下，我们证明界面的厚度恰好是$ \ varepsilon $阶，这是最佳厚度估计。我们注意到，由于存在非局部项，因此比较原理不适用于我们的方程式，因此早期标准Allen-Cahn方程研究中使用的方法不起作用。因此，我们将采取不同的方法，