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

## TL;DR

This paper provides precise criteria based on initial mass distribution for the instantaneous movement of free boundaries in solutions to the thin-film equation, advancing understanding of free boundary behavior.

## Contribution

It introduces sharp, necessary and sufficient criteria for free boundary propagation in the thin-film equation based on initial mass distribution, using novel propagation of degeneracy techniques.

## Key findings

- Criteria are necessary and sufficient in the weak slippage regime.
- Propagation of degeneracy strategy links local mass and energy estimates.
- New estimate connects mass motion to entropy production.

## Abstract

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