Stability of Planar Waves in the Allen–Cahn Equation

Abstract

We study the asymptotic stability of planar waves for the Allen-Cahn equation on n , 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 n . 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.

Stability of planar waves in mono-stable reaction-diffusion equations

“…This is similar to that of [7], and therefore we omit it here. Theorem 1.2 and Proposition 1.1 can be proved by constructing the suitable upper and lower solutions, which are similar to those of [7]. We omit the details here.…”

“…Similar to [7], we have the following proposition, which shows that the convergence rate (1.7) is optimal in some sense. Proposition 1.1.…”

“…When the reaction term f (u) is of the bistable type, this problem has been studied in [7]. They obtained the stability of planar waves under any, possibly large, initial perturbations and almost periodic perturbations.…”

“…Encouraged by [7], in this paper, we will consider equations (1.1) with (1.2) in R n with n ≥ 2. Throughout this paper, we always assume that φ is a monotone solution of (1.4) and satisfies the fact that φ(0) = 1/2.…”

Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation

“…One interesting avenue of future work is to investigate the multidimensional stability of planar traveling waves for equation (1.1) under weaker assumptions for the perturbations. For example, in the local case, Matano et al have recently shown that [15] the multidimensional stability of planar traveling waves with possibly large initial perturbations that only decay at space infinity. It would be interesting to see if their techniques can adapted to our nonlocal setting.…”