Slow motion for one-dimensional nonlinear damped hyperbolic Allen-Cahn systems

TL;DR

This paper investigates the slow evolution of metastable states in a one-dimensional nonlinear damped hyperbolic reaction-diffusion system, extending previous results for the Allen-Cahn equation to systems.

Contribution

It extends the energy approach to analyze metastable dynamics from the Allen-Cahn equation to hyperbolic systems, proving existence and persistence of metastable states with exponentially slow layer motion.

Findings

Metastable states persist for exponentially long times as diffusion coefficient tends to zero.

Transition layers move with exponentially small velocity.

The approach generalizes previous methods to hyperbolic reaction-diffusion systems.

Abstract

We consider a nonlinear damped hyperbolic reaction-diffusion system in a bounded interval of the real line with homogeneous Neumann boundary conditions and we study the metastable dynamics of the solutions. Using an "energy approach" introduced by Bronsard and Kohn [CPAM 1990] to study slow motion for Allen-Cahn equation and improved by Grant [SIAM J. Math. Anal. 1995] in the study of Cahn-Morral systems, we improve and extend to the case of systems the results valid for the hyperbolic Allen-Cahn equation. In particular, we study the limiting behavior of the solutions as $\varepsilon\to0^+$, where $\varepsilon^2$ is the diffusion coefficient, and we prove existence and persistence of metastable states for a time $T_\varepsilon>\exp(A/\varepsilon)$. Such metastable states have a transition layer structure and the transition layers move with exponentially small velocity.