On the Metastability of the 1-d Allen–Cahn Equation

We apply an energy method for metastability, developed in an earlier work with Otto, to the Allen–Cahn equation on the line and a broad class of initial data. In the earlier work, we for simplicity considered the equation on (0,L)×(0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0,L)\times (0,\infty )$$\end{document} with periodic boundary conditions and an initial condition with two simple zeros. In this paper we explain the implications of the metastability framework (slightly modified as in previous joint work with Scholtes) for the equation on R×(0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}\times (0,\infty )$$\end{document} and a more general initial condition. Our goal is to make clear the strength of the metastability framework and to highlight the difference in the analysis between the second-order Allen–Cahn equation and the fourth-order Cahn–Hilliard equation.