Heteroclinic traveling waves of 2D parabolic Allen-Cahn systems

n this paper we show the existence of traveling waves $w: [0,+\infty) \times \mathbb{R}^2 \to \mathbb{R}^k$ ($k \geq 2$) for the parabolic Allen-Cahn system \begin{equation} \partial_t w - \Delta w = -\nabla_u V(w) \mbox{ in } [0,+\infty) \times \mathbb{R}^2, \end{equation} satisfying some \textit{heteroclinic} conditions at infinity. The potential $V$ is a non-negative 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 1D heteroclinics which have different energies and the speed $c^\star$ satisfies certain uniqueness properties. The proof of variational and, in particular, it requires the assumption of an upper bound, depending on $V$, on the difference between the energies of the 1D heteroclinics.