Droplet spreading under weak slippage: the optimal asymptotic propagation rate in the multi-dimensional case

We prove optimal estimates on the growth rate of the support of solutions to the thin-film equation ut + div(|u|n∇∆u) = 0 in space dimensions N = 2 and N = 3 for parameters n ∈ [2, 3) which correspond to Navier’s slip condition (n = 2) or certain variants modeling weaker slippage effects. Our approach relies on a new class of weighted energy estimates. It is inspired by the onedimensional technique of Hulshof and Shishkov Adv. Diff. Equations 3, (1998) 625–642, and it simplifies their method, mainly with respect to basic integral estimates to be used.