Metastable dynamics for a hyperbolic variant of the mass conserving Allen-Cahn equation in one space dimension. (English) Zbl 1456.35011
Summary: In this paper, we consider some hyperbolic variants of the mass conserving Allen-Cahn equation, which is a nonlocal reaction-diffusion equation, introduced (as a simpler alternative to the Cahn-Hilliard equation) to describe phase separation in binary mixtures. In particular, we focus our attention on the metastable dynamics of solutions to the equation in a bounded interval of the real line with homogeneous Neumann boundary conditions. It is shown that the evolution of profiles with \(N + 1\) transition layers is very slow in time and we derive a system of ODEs, which describes the exponentially slow motion of the layers. A comparison with the classical Allen-Cahn and Cahn-Hilliard equations and theirs hyperbolic variations is also performed.
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
| 35L71 | Second-order semilinear hyperbolic equations |
| 35R09 | Integro-partial differential equations |
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