Optimal lower bounds on asymptotic support propagation rates for the thin-film equation

We derive lower bounds on asymptotic support propagation rates for strong solutions of the Cauchy problem for the thin-film equation. The bounds coincide up to a constant factor with the previously known upper bounds and thus are sharp. Our results hold in case of at most three spatial dimensions and n∈(1,2.92)n∈(1,2.92). The result is established using weighted backward entropy inequalities with singular weight functions to yield a differential inequality; combined with some entropy production estimates, the optimal rate of propagation is obtained. To the best of our knowledge, these are the first lower bounds on asymptotic support propagation rates for higher-order nonnegativity-preserving parabolic equations.