Explicit solutions of a two-dimensional fourth-order nonlinear diffusion equation

We construct two-dimensional polynomial solutions of the fourth-order nonlinear diffusion equation ut+∇·(u∇∇2u)=0. The interfaces and contour lines are given by quadratic curves: ellipses, hyperbolas, and parabolas. The solution with elliptical interfaces represents the spreading of an scalar initially concentrated along a segment. For the cases of the parabolic and hyperbolic interfaces, self-similar solutions are obtained. These solutions may be extended to three and higher dimensions. Some other extensions are also noted.