Touchdown-singularity formation and criticality in the thin-film equation

The thin-film equation (TFE), [Formula: see text], is significant physically in the description of surface-tension-driven flows of thin films of viscous liquids and has served an important role mathematically in elucidating the properties and challenges of high-order degenerate parabolic equations. Long-standing open questions nevertheless remain, of which perhaps the most important is the identification of the critical value of the exponent [Formula: see text] above which film rupture is not possible. Here, we apply a combination of analytical and numerical methods to further the understanding of this issue, uncovering new types of touchdown behaviour that lead to concrete conjectures regarding the role of [Formula: see text] in this type of criticality.This article is part of the theme issue 'Science into the next millennium: 25 years on'.