On asymptotic stability in 3D of kinks for the $\phi ^4$ model

TL;DR

This paper proves the asymptotic stability of 1D kinks extended into 3D under the nonlinear wave equation, using separation of variables and dispersion techniques from Klein-Gordon and Schrödinger equations.

Contribution

It extends the stability analysis of kinks from 1D to 3D by combining methods from reaction diffusion, Klein-Gordon, and Schrödinger equations.

Findings

Proves asymptotic stability of 3D kinks under NLW

Uses separation of variables for transversal and longitudinal directions

Employs dispersion results for Schrödinger operators in 1D

Abstract

We add to a kink, which is a 1 dimensional structure, two transversal directions. We then check its asymptotic stability with respect to compactly supported perturbations in 3D and a time evolution under a Nonlinear Wave Equation (NLW). The problem is inspired by work by Jack Xin on asymptotic stability in dimension larger than 1 of fronts for reaction diffusion equations. The proof involves a separation of variables. The transversal variables are treated as in work on Nonlinear Klein Gordon Equation (NLKG) originating from Klainerman and from Shatah in a particular elaboration due to Delort and others. The longitudinal variable is treated by means of a result by Weder on dispersion for Schroedinger operators in 1D.