On asymptotic stability in 3D of kinks for the 𝜙⁴ model

Cuccagna, Scipio

doi:10.1090/S0002-9947-07-04356-5

On asymptotic stability in 3D of kinks for the $\phi ^4$ model
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Trans. Amer. Math. Soc. 360 (2008), 2581-2614 Request permission

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 et al. The longitudinal variable is treated by means of a result by Weder on dispersion for Schrödinger operators in 1D.

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