  The Fourth order $\phi^4$ model generalizes the classical $\phi^4$ model of
quantum field theory, sharing the same kink solution. It is also the dispersive
counterpart of the well-known parabolic Cahn-Hilliard equation. Mathematically
speaking, the kink is characterized by a fourth-order nonnegative linear
operator with a simple kernel at the origin but no spectral gap. In this paper,
we consider the kink of this theory, and prove orbital and asymptotic stability
for any perturbation in the energy space.
