On asymptotic stability in 3D of kinks for the $\phi ^4$ model
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.
Keywords
Cite
@article{arxiv.0801.2678,
title = {On asymptotic stability in 3D of kinks for the $\phi ^4$ model},
author = {Scipio Cuccagna},
journal= {arXiv preprint arXiv:0801.2678},
year = {2008}
}
Comments
To appear on Transactions of the American Mathematical Society