Stable patterns for fourth-order parabolic equations

Abstract

We consider fourth-order parabolic equations of gradient type. For the sake of simplicity, the analysis is carried out for the specific equation $u\sb t=-\gamma\ u\sb {xxxx}+\beta u\sb {xx}-F\sp \prime(u)$ with $(t,x)\in (0,\infty)\times(0, L)$ and $\gamma,\beta>0$ and where $F(u)$ is a bistable potential. We study its stable equilibria as a function of the ratio $\gamma/beta\sp 2$. As the ratio $\gamma/beta\sp 2$ crosses an explicit threshold value, the number of stable patterns grows to infinity as $L\to \infty$. The construction of the stable patterns is based on a variational gluing method that does not require any genericity conditions to be satisfied.