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 [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 $\epsilon \to 0^{+}$, where ${\epsilon }^2$ is the diffusion coefficient, and we prove existence and persistence of metastable states for a time $T_{\epsilon }>\exp (A/\epsilon )$. Such metastable states have a transition layer structure and the transition layers move with exponentially small velocity.

24 pages

References in corpus (1)

• Slow motion for a hyperbolic variation of Allen-Cahn equation in one space dimension

Cited by in corpus (4)

• Slow dynamics for the hyperbolic Cahn-Hilliard equation in one space dimension
• Slow motion for a hyperbolic variation of Allen-Cahn equation in one space dimension
• Exponentially slow motion of interface layers for the one-dimensional Allen-Cahn equation with nonlinear phase-dependent diffusivity
• Motion of interfaces for a damped hyperbolic Allen-Cahn equation