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

Abstract

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