Masaharu Taniguchi

TRAVELING FRONTS OF PYRAMIDAL SHAPES IN THE ALLEN-CAHN EQUATIONS

This paper studies pyramidal traveling fronts in the Allen–Cahn equation or in the Nagumo equation. For the nonlinearity we are concerned mainly with the bistable reaction term with unbalanced energy density. Two-dimensional V-form waves and cylindrically symmetric waves in higher dimensions have been recently studied. Our aim in this paper is to construct truly three-dimensional traveling waves. For a pyramid that satisfies a condition, we construct a traveling front for which the contour line has a pyramidal shape. We also construct generalized pyramidal fronts and traveling waves of a hybrid type between pyramidal waves and planar V-form waves. We use the comparison principles and construct traveling fronts between supersolutions and subsolutions.

Stability of Planar Waves in the Allen–Cahn Equation

We study the asymptotic stability of planar waves for the Allen–Cahn equation on ℝ 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 ℝ 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.

Multi-dimensional pyramidal travelling fronts in the Allen-Cahn equations

We study travelling-front solutions of pyramidal shapes in the Allen–Cahn equation in RN with N 3. It is well known that two-dimensional V-form travelling fronts and three-dimensional pyramidal travelling fronts exist and are stable. The aim of this paper is to show that for N 4 there exist N -dimensional pyramidal travelling fronts. We construct a supersolution and a subsolution, and find a pyramidal travelling-front solution between them. For the construction of a supersolution we use a multi-scale method.

Traveling fronts of pyramidal shapes in competition-diffusion systems

It is well known that a competition-diffusion system has a one-dimensional traveling front. This paper studies traveling front solutions of pyramidal shapes in a competition-diffusion system in

Instability of planar interfaces in reaction-diffusion systems

\mathbb{R}^N

An (N-1)-dimensional convex compact set gives an N-dimensional traveling front in the Allen--Cahn equation

with

Existence and global stability of traveling curved fronts in the Allen-Cahn equations

N\geq 2

Global stability of traveling curved fronts in the Allen-Cahn equations

. By using a multi-scale method, we construct a suitable pair of a supersolution and a subsolution, and find a pyramidal traveling front solution between them.

The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen–Cahn equations

Instability of planar front solutions to reaction-diffusion systems in two space dimensions is studied. Let

Multi-dimensional traveling fronts in bistable reaction-diffusion equations

\varepsilon