2021/06/17 by Ramon Oliver-Bonafoux, Oliver-Bonafoux, Ramon
Mathematics · #35J20 #35K40 #49J27 #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2106.09441
openalex publication_date 2021/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
n this paper we show the existence of traveling waves w: [0,+∞) × ℝ2 → ℝk (k ≥ 2) for the parabolic Allen-Cahn system ∂t w - Δw = -∇u V(w) in [0,+∞) × ℝ2, satisfying some 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^⋆>0 and a profile \mathfrakU. The profile \mathfrakU joins as x1 → ± ∞ (in a suitable sense) two locally minimizing 1D heteroclinics which have different energies and the speed c^⋆ 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.