We prove well-posedness, partial regularity, and stability of the thin-film equation h_t+(m(h)h_zzz)_z=0 with general mobility m(h)=hⁿ and mobility exponent n∈(1,3/2)∪(3/2,3) in the regime of perfect wetting (zero contact angle). After a suitable coordinate transformation to fix the free boundary (the contact line where liquid, air, and solid coalesce), the thin-film equation is rewritten as an abstract Cauchy problem and we obtain maximal L_t^p-regularity for the linearized evolution. Partial regularity close to the free boundary is obtained by studying the elliptic regularity of the spatial part of the linearization. This yields solutions that are non-smooth in the distance to the free boundary, in line with previous findings for source-type self-similar solutions. In a scaling-wise quasi-minimal norm for the initial data, we obtain a well-posedness and asymptotic stability result for perturbations of traveling waves. The novelty of this work lies in the usage of L^p-estimates in time, where 1<p<∞, while the existing literature mostly deals with p=2 at least for nonlinear mobilities. This turns out to be essential to obtain for the first time a well-posedness result in the perfect-wetting regime for all physical nonlinear slip conditions except for a strongly degenerate case at n=3/2 and the well-understood Greenspan-slip case n=1. Furthermore, compared to [36] by Giacomelli, the first author of this paper, Knüpfer, and Otto, where a PDE approach yields L_t²-estimates, well-posedness, and stability for 1.8384≈3/17(15−√21)<n<3/11(7+√5)≈2.5189, our functional-analytic approach is shorter while at the same time giving a more general result.

We consider the thin-film equation ∂th+∂ym(h)∂y3h=0 in {h > 0} with partial-wetting boundary conditions and inhomogeneous mobility of the form m(h) = h 3 + λ 3-n h n , where h ∼ 0 is the film height, λ > 0 is the slip length, y > 0 denotes the lateral variable, and n ϵ (0, 3) is the mobility exponent parameterizing the nonlinear slip condition. The partial-wetting regime implies the boundary condition ∂ y h = const. > 0 at the triple junction ∂{h > 0} (nonzero microscopic contact angle). Existence and uniqueness of traveling-wave solutions to this problem under the constraint ∂y2h→0 as h → ∞ have been proved in previous work by Chiricotto and Giacomelli (2011 Commun. Appl. Ind. Math. 2 e-388, 16). We are interested in the asymptotics as h ↓ 0 and h → ∞. By reformulating the problem as h ↓ 0 as a dynamical system for the difference between the solution and the microscopic contact angle, values for n are found for which linear as well as nonlinear resonances occur. These resonances lead to a different asymptotic behavior of the solution as h ↓ 0 depending on n. Together with the asymptotics as h → ∞ characterizing the Cox-Voinov law for the velocity-dependent macroscopic contact angle as found by Giacomelli, the first author of this work, and Otto (2016 Nonlinearity 29 2497-536), the rigorous asymptotics of traveling-wave solutions to the thin-film equation in partial wetting can be characterized. Furthermore, our approach enables us to analyze the relation between the microscopic and macroscopic contact angle. It is found that the Cox-Voinov law for the macroscopic contact angle depends continuously differentiably on the microscopic contact angle.