Metastable dynamics for hyperbolic variations of the Allen-Cahn equation

Abstract

Metastable dynamics of a hyperbolic variation of the Allen-Cahn equation with homogeneous Neumann boundary conditions are considered. Using the "dynamical approach" proposed by Carr-Pego to study slow-evolution of solutions in the classic parabolic case, we prove existence and persistence of metastable patterns for an exponentially long time. In particular, we show the existence of an "approximately invariant" N -dimensional manifold M 0 for the hyperbolic Allen-Cahn equation: if the initial datum is in a tubular neighborhood of M 0 , the solution remains in such neighborhood for an exponentially long time. Moreover, the solution has N transition layers and the transition points move with exponentially small velocity. In addition, we determine the explicit form of a system of ordinary differential equations describing the motion of the transition layers and we analyze the differences with the corresponding motion valid for the parabolic case.

Slow motion for a hyperbolic variation of Allen–Cahn equation in one space dimension

“…This approach can be refined to obtain persistence of the layered structure for time intervals of O(e C/ε ) and extended to the case when u is vector-valued. These topics, as well as the study of metastability for hyperbolic Allen-Cahn equation with the dynamical approach of Carr and Pego, are addressed in [12,13].…”