This paper is concerned with existence and stability of traveling curved fronts for the Allen-Cahn equation in the two-dimensional space. By using the supersolution and the subsolution, we construct a traveling curved front, and show that it is the unique traveling wave solution between them. Our supersolution can be taken arbitrarily large, which implies some global asymptotic stability for the traveling curved front.

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.

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 threedimensional 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.

This paper studies the uniqueness and the asymptotic stability of a pyramidal traveling front in the three-dimensional whole space. For a given admissible pyramid we prove that a pyramidal traveling front is uniquely determined and that it is asymptotically stable under the condition that given perturbations decay at infinity. For this purpose we characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces. Moreover we characterize the pyramidal traveling front in another way, that is, we write it as a combination of two-dimensional V-form waves on the edges. This characterization also uniquely determines a pyramidal traveling front.

Abstract. This paper is concerned with asymptotical stability of traveling curved fronts to a mean curvature flow with a constant driving force term in the two-dimensional Euclidean space. Our first result shows that, if a suitable bounded perturbation is added to a traveling curved front, it does not recover its shape at any positive time. This fact implies that boundedness of given perturbations is not enough for asymptotical stability. Then we prove that, if a given bounded perturbation decays at infinity, the perturbed traveling curved fronts always recover their shapes as time goes on. This fact holds true for any large perturbation if it decays at infinity. Introduction.Pattern formation is one of the most attractive fields in applied mathematics. Since traveling waves play important roles on the pattern formation, many researchers have studied them. Traveling waves in an one-dimensional media or planar traveling fronts in the plane seem to be mainly investigated so far. Fife [6] studied corner layers in the Allen-Cahn dynamics, which gives a first step to study traveling waves with more complicated shapes.This paper is concerned with the asymptotical stability of traveling curved fronts for a curvature flow with a constant driving force term in R 2 . Let Define r(t) = dD(i). Let 1/ be the normal vector on r(t) pointing from D(t) to D(t)c . The curvature H is given by H = -divi*. A pair of