Solutions with single radial interface of the generalized Cahn-Hilliard flow

We consider the generalized parabolic Cahn-Hilliard equation $$u_t=-\mathrm{\Delta }[\mathrm{\Delta }u-W^{\mathrm{\prime }}(u)]+W^{\mathrm{\prime }\mathrm{\prime }}(u)[\mathrm{\Delta }u-W^{\mathrm{\prime }}(u)]\mathrm{\forall }(t,x)\in \tilde{\mathbb{R}}\times {\mathbb{R}}^n,$$ where $n=2$ or $n\ge 4$, $W(\cdot )$ is the typical double-well potential function and $\tilde{\mathbb{R}}$ is given by $$\tilde{\mathbb{R}}=\{\begin{array}{rll}(0,\mathrm{\infty }),& ifn=2,(-\mathrm{\infty },0),& \text{quadif}n\ge 4.\end{array}$$ 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 $\omega (\mathrm{\mid }x\mathrm{\mid })$, which satisfies ${\omega }^{\mathrm{\prime }\mathrm{\prime }}(y)-W^{\mathrm{\prime }}(\omega (y))=0$. Its interface is resemble at main order copy of the sphere of the following form $$\mathrm{\mid }x\mathrm{\mid }=\sqrt[4]{-2(n-3)(n-1{)}^{2}t},\mathrm{\forall }(t,x)\in \tilde{\mathbb{R}}\times {\mathbb{R}}^n,$$ which is a solution to the Willmore flow in Differential Geometry. When $n=1$ or $3$, the result consists trivial solutions.