Heteroclinic traveling waves of two-dimensional parabolic Allen–Cahn systems

Abstract

In this paper we show the existence of traveling waves $w:[0,+\infty )\times {\mathbb{R}}^2\to {\mathbb{R}}^k$ ($k\ge 2$) for the parabolic Allen–Cahn system ${\partial }_{t}w-\Delta w=-{\nabla }_{u}V(w)$ in $[0,+\infty )\times {\mathbb{R}}^2$, satisfying some heteroclinic conditions at infinity. The potential $V$ is a nonnegative and smooth multi-well potential, which means that its null set is finite and contains at least two elements. The traveling wave $w$ propagates along the horizontal axis according to a speed $c^{\star }>0$ and a profile $\mathfrak{U}$. The profile $\mathfrak{U}$ joins as $x_1\to \pm \infty$ (in a suitable sense) two locally minimizing one-dimensional heteroclinics which have different energies, and the speed $c^{\star }$ satisfies certain uniqueness properties. The proof is variational and, in particular, it requires the assumption of an upper bound, depending on $V$, on the difference between the energies of the one-dimensional heteroclinics.