Interface dynamics in semilinear wave equations

Abstract

We consider the wave equation epsilon(2)(-partial derivative(2)(t) + Delta)u + f(u) = 0 for 0 < epsilon << 1, where f is the derivative of a balanced, double-well potential, the model case being f(u) = u - u(3). For equations of this form, we construct solutions that exhibit an interface of thickness O(epsilon) that separates regions where the solution is O(epsilon(k)) close to +/- 1, for k >= 1, and that is close to a timelike hypersurface of vanishing Minkowskian mean curvature. This provides a Minkowskian analog of the numerous results that connect the Euclidean Allen-Cahn equation and minimal surfaces or the parabolic Allen-Cahn equation and motion by mean curvature. Compared to earlier results of the same character, we develop a new constructive approach that applies to a larger class of nonlinearities and yields much more precise information about the solutions under consideration.