Slow motion for one-dimensional nonlinear damped hyperbolic Allen-Cahn systems
Texas State University, Department of Mathematics
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  to study slow motion for Allen-Cahn equation and improved by Grant  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 ). 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.
Hyperbolic reaction-diffusion systems, Allen-Cahn equation, Metastability, Energy estimates
Folino, R. (2019). Slow motion for one-dimensional nonlinear damped hyperbolic Allen-Cahn systems. <i>Electronic Journal of Differential Equations, 2019</i>(113), pp. 1-21.