Symmetric Singularity Formation In Lubrication-Type Equations For Interface Motion

. Fourth order degenerate diffusion equations arise in a `lubrication approximation' of a thin film or neck driven by surface tension. Numerical studies of the lubrication equation (LE) h t + (h n hxxx )x = 0 with various boundary conditions indicate that singularity formation in which h(x(t); t) ! 0 occurs for small enough n with `anomalous' or `second type' scaling inconsistent with usual dimensional analysis. This paper considers locally symmetric or even singularities in the (LE) and in the modified lubrication equation (MLE) h t + h n hxxxx = 0. Both equations have the property that entropy bounds forbid finite time singularities when n is sufficiently large. Power series expansions for local symmetric similarity solutions are proposed for equation (LE) with n ! 1 and (MLE) for all n 2 R. In the latter case, special boundary conditions that force singularity formation are required to produce singularities when n is large. Matching conditions at higher order terms in the expa..