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We establish sharp criteria for the instantaneous propagation of freeboundaries in solutions to the thin-film equation. The criteria are formulatedin terms of the initial distribution of mass (as opposed to previousalmost-optimal results), reflecting the fact that mass is a locally conservedquantity for the thin-film equation. In the regime of weak slippage, ourcriteria are at the same time necessary and sufficient. The proof of our upperbounds on free boundary propagation is based on a strategy of "propagation ofdegeneracy" down to arbitrarily small spatial scales: We combine estimates onthe local mass and estimates on energies to show that "degeneracy" on a certainspace-time cylinder entails "degeneracy" on a spatially smaller space-timecylinder with the same time horizon. The derivation of our lower bounds on freeboundary propagation is based on a combination of a monotone quantity andalmost optimal estimates established previously by the second author with a newestimate connecting motion of mass to entropy production.

Sharp criteria for the waiting time phenomenon in solutions to the thin-film equation