Slow motion for one-dimensional nonlinear damped hyperbolic Allen-Cahn 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 [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 ε → 0+, where ε2 is the diffusion coefficient, and we prove existence and persistence of metastable states for a time Tε > exp(A/ε). Such metastable states have a transition layer structure and the transition layers move with exponentially small velocity.