Stability of planar waves in the Allen-Cahn equation

We study the asymptotic stability of planar waves for the Allen-Cahn equation on ℝⁿ, where n ≥ 2. Our first result states that planar waves are asymptotically stable under any-possibly large-initial perturbations that decay at space infinity. Our second result states that the planar waves are asymptotically stable under almost periodic perturbations. More precisely, the perturbed solution converges to a planar wave as t → ∞. The convergence is uniform in ℝⁿ. Lastly, the existence of a solution that oscillates permanently between two planar waves is shown, which implies that planar waves are not asymptotically stable under more general perturbations.