Heteroclinic traveling waves of two-dimensional parabolic Allen–Cahn systems
Ramon Oliver-Bonafoux
Università di Verona, Verona, Italy
![Heteroclinic traveling waves of two-dimensional parabolic Allen–Cahn systems cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserials%2Fcover-aihpc.png&w=3840&q=90)
Abstract
In this paper we show the existence of traveling waves () for the parabolic Allen–Cahn system in , satisfying some heteroclinic conditions at infinity. The potential 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 propagates along the horizontal axis according to a speed and a profile . The profile joins as (in a suitable sense) two locally minimizing one-dimensional heteroclinics which have different energies, and the speed satisfies certain uniqueness properties. The proof is variational and, in particular, it requires the assumption of an upper bound, depending on , on the difference between the energies of the one-dimensional heteroclinics.
Cite this article
Ramon Oliver-Bonafoux, Heteroclinic traveling waves of two-dimensional parabolic Allen–Cahn systems. Ann. Inst. H. Poincaré C Anal. Non Linéaire (2024), published online first
DOI 10.4171/AIHPC/127