Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 113, pp. 1-21.

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

Author: Raffaele Folino (Univ. degli Studi dell'Aquila, Italy)

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 [11]
 to study slow motion for Allen-Cahn equation and improved by Grant [25]
 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 (see [18]).
 In particular, we study the limiting behavior of the solutions as
 $\varepsilon\to 0^+$, 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.

Submitted March 30, 2019. Published October 2, 2019.
Math Subject Classifications: 35L53, 35B25, 35K57.
Key Words: Hyperbolic reaction-diffusion systems; Allen-Cahn equation;
           metastability; energy estimates.