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.