  We study a higher order parabolic partial differential equation that arises
in the context of condensed matter physics. It is a fourth order semilinear
equation whose nonlinearity is the determinant of the Hessian matrix of the
solution. We consider this model in a bounded domain of the real plane and
study its stationary solutions both when the geometry of this domain is
arbitrary and when it is the unit ball and the solution is radially symmetric.
We also consider the initial-boundary value problem for the full parabolic
equation. We summarize our results on existence of solutions in these cases and
propose an open problem related to the existence of self-similar solutions.
