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

Abstract

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.

Existence and qualitative properties of multidimensional conical bistable fronts

“…The proofs can be carried over word by word in the bistable case, which is actually simpler. See also [23]. The plan of the paper is the following.…”

“…A part of this result is proved in [23], by the construction of a super-solution coming from the study of travelling waves for the 2D mean curvature motion with drift.…”

An $(N-1)$-Dimensional Convex Compact Set Gives an $N$-Dimensional Traveling Front in the Allen--Cahn Equation

“…For the Allen-Cahn equation, multidimensional traveling fronts have been studied by many mathematicians. Two-dimensional V-form fronts are studied by Ninomiya and myself [13,14], Hamel, Monneau, and Roquejoffre [6,7], Haragus and Scheel [8], and so on. Cylindrically symmetric traveling fronts in R N are studied by [6,7].…”

Symmetry of Traveling Wave Solutions to the Allen–Cahn Equation in $${{\mathbb R^{2}}}$$

“…The difference of zero speed and positive speed of one dimensional traveling wave solution g for the balanced and unbalanced potential leads to a fundamental difference of the structure of traveling fronts in higher dimensional spaces, as discussed below, as well as shown in Theorem 1.1 and Theorem 3.2 and [24]. The existence, uniqueness, stability and other qualitative properties of traveling wave solutions to the unbalanced AllenCahn equation have been studied in [30] [31], [42], [43], [45], [46]. Similar traveling wave solutions for Fisher-KPP type equation or combustion equation have also been investigated in [11], [28], [32], [41].…”