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We prove well-posedness, partial regularity, and stability of a thin-filmequation $h_t + (m(h) h_{zzz})_z = 0$ with general mobility $m(h) = h^n$ andmobility exponent $n\in (1,\tfrac{3}{2})\cup (\tfrac{3}{2},3)$ in the regime ofperfect wetting (zero contact angle). After a suitable coordinatetransformation to fix the free boundary (the contact line where liquid, air,and solid coalesce), the thin-film equation is rewritten as an abstract Cauchyproblem and we obtain maximal $L^{p}_t$-regularity for the linearizedevolution. Partial regularity close to the free boundary is obtained bystudying the elliptic regularity of the spatial part of the linearization. Thisyields solutions that are non-smooth in the distance to the free boundary, inline with previous findings for source-type self-similar solutions. In ascaling-wise quasi-minimal norm for the initial data, we obtain awell-posedness and asymptotic stability result for perturbations of travelingwaves. The novelty of this work lies in the usage of $L^{p}$ estimates in time,where $1 < p < \infty$, while the existing literature mostly deals with $p = 2$at least for nonlinear mobilities. This turns out to be essential to obtain forthe first time a well-posedness result in the perfect-wetting regime for allphysical nonlinear slip conditions except for a strongly degenerate case at $n= \tfrac 3 2$ and the well-understood perfect-slip case $n = 1$. Furthermore,compared to [J. Differential Equations, 257(1):15-81, 2014] by Giacomelli, thefirst author of this paper, Kn\"upfer, and Otto, where a PDE-approach yields$L^2_t$-estimates, well-posedness, and stability for $1.8384 \approx\tfrac{3}{17}(15-\sqrt{21}) < n < \tfrac{3}{11}(7+\sqrt{5}) \approx 2.5189$,our functional-analytic approach is significantly shorter while at the sametime giving a more general result.

Classical solutions to the thin-film equation with general mobility in the perfect-wetting regime