This paper considers a one-dimensional generalized Allen–Cahn equation of the form

where \(\varepsilon > 0\) is constant, \(D = D(u)\) is a positive, uniformly bounded below, diffusivity coefficient that depends on the phase field u, and f(u) is a reaction function that can be derived from a double-well potential with minima at two pure phases \(u = \alpha \) and \(u = \beta \). It is shown that interface layers (namely, solutions that are equal to \(\alpha \) or \(\beta \) except at a finite number of thin transitions of width \(\varepsilon \)) persist for an exponentially long time proportional to \(\exp (C/\varepsilon )\), where \(C > 0\) is a constant. In other words, the emergence and persistence of metastable patterns for this class of equations is established. For that purpose, we prove energy bounds for a renormalized effective energy potential of Ginzburg–Landau type. Numerical simulations, which confirm the analytical results, are also provided.

本文考虑了形式为一维的广义Allen-Cahn方程

其中\（\ varepsilon> 0 \）是常数，\（D = D（u）\）是一个正数，在下面均匀有界，扩散系数取决于相场u，而f（u）是一个反应函数，可以从具有两个纯相位\（u = \ alpha \）和\（u = \ beta \）的最小值的双阱势中得出。结果表明，接口层（即，在有限数量的宽度\（\ varepsilon \）的细过渡处除外，等于\（\ alpha \）或\（\ beta \）的解决方案）持续了指数级的长时间正比于\（\ EXP（C / \ varepsilon）\） ，其中\（C> 0 \）是一个常数。换句话说，建立了此类方程的亚稳态模式的出现和持久性。为此，我们证明了Ginzburg-Landau类型的重新规范化的有效能势的能界。还提供了数值模拟，证实了分析结果。