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

Abstract

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.