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

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