Metastable dynamics for hyperbolic variations of the Allen-Cahn equation

TL;DR

This paper investigates the metastable behavior of a hyperbolic version of the Allen-Cahn equation, demonstrating the existence of long-lasting metastable patterns, an invariant manifold, and deriving the dynamics of transition layers.

Contribution

It extends the dynamical approach to metastability from the parabolic to the hyperbolic Allen-Cahn equation, establishing long-term pattern persistence and explicit layer dynamics.

Findings

Existence of an approximately invariant manifold for the hyperbolic Allen-Cahn equation.

Solutions with initial data near this manifold stay close for exponentially long times.

Explicit ODE system describing the motion of transition layers and comparison with the parabolic case.

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 [10] and Fusco-Hale [19] 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 $\mathcal{M}_0$ for the hyperbolic Allen-Cahn equation: if the initial datum is in a tubular neighborhood of $\mathcal{M}_0$, 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.