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

Details On asymptotic stability in 3D of kinks for the $φ^4$ model On asymptotic stability in 3D of kinks for the $φ^4$ model

2008-01-17

arxiv.org/abs/0801.2678v1

Mathematics

Analysis of PDEs

To appear on Transactions of the American Mathematical Society

Scientific paper

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.

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