  We consider the generalized parabolic Cahn-Hilliard equation $$
u_t=-\Delta\left[\Delta u -W'(u)\right]+W''(u)\left[\Delta u -W'(u)\right]
\qquad \forall\, (t, x)\in \widetilde{{\mathbb R}}\times{\mathbb R}^n, $$ where
$n=2$ or $n\geq 4$, $W(\cdot)$ is the typical double-well potential function
and $\widetilde{\mathbb R}$ is given by $$ \widetilde{\mathbb R}=\left\{
  \begin{array}{rl}
  (0, \infty), &\quad \mbox{if } n=2,
  (-\infty, 0), & \quad\mbox{if } n\geq 4.
  \end{array}
  \right. $$ We construct a radial solution $u(t, x)$ possessing an interface.
At main order this solution consists of a traveling copy of the steady state
$\omega(|x|)$, which satisfies $\omega''(y)-W'(\omega(y))=0$. Its interface is
resemble at main order copy of the sphere of the following form $$
|x|=\sqrt[4]{-2(n-3)(n-1)^2t}, \qquad \forall\, (t, x)\in \widetilde{{\mathbb
R}}\times{\mathbb R}^n, $$ which is a solution to the Willmore flow in
Differential Geometry. When $n=1$ or $3$, the result consists trivial
solutions.
