  The asymptotic properties of the stability potentials of kinks with power-law
tails are discussed. In particular, in cosmology such kinks can describe
"thick" domain walls. The discrete part of the domain wall excitation spectrum,
the profile of which is described by a kink solution with one or both power-law
asymptotics, cannot contain levels other than zero (translational) mode.
Nevertheless, it can be shown that scenarios are quite possible when long-lived
localized vibrations will be excited on the domain wall. This, in turn, can
affect the processes of interaction of two or more domain walls.
