Iron, DavidKolokolonikov, TheodoreRumsey, JohnWei, Juncheng2013-10-042013-10-042009Iron, David, Theodore Kolokolonikov, John Rumsey, and Juncheng Wei. 2009. "Stability of Curved Interfaces in the Perturbed Two-Dimensional Allen-Cahn System." SIAM Journal on Applied Mathematics 69(5): 1228-16.00361399http://dx.doi.org/10.1137/070706380http://hdl.handle.net/10222/37345We consider the singular limit of a perturbed Allen-Cahn model on a bounded two-dimensional domain: $\left\{\begin{array}{@{}ll@{}} u_t = \varepsilon^2 \Delta u - 2 (u - \varepsilon a) (u^2 - 1), & x \in \Omega \subset \mathbb{R}^2 \ \partial_n u = 0, & x \in \partial \Omega \end{array} \right.$ where $\varepsilon$ is a small parameter and $a$ is an $O(1)$ quantity. We study equilibrium solutions that have the form of a curved interface. Using singular perturbation techniques, we fully characterize the stability of such an equilibrium in terms of a certain geometric eigenvalue problem, and give a simple geometric interpretation of our stability results. Full numerical computations of the time-dependent PDE as well as of the associated two-dimensional eigenvalue problem are shown to be in excellent agreement with the analytical predictions.enText6951228