  We consider the thin-film equation $\partial_t h + \partial_y \left(m(h)
\partial_y^3 h\right) = 0$ in $\{h > 0\}$ with partial-wetting boundary
conditions and inhomogeneous mobility of the form $m(h) =
h^3+\lambda^{3-n}h^n$, where $h \ge 0$ is the film height, $\lambda > 0$ is the
slip length, $y > 0$ denotes the lateral variable, and $n \in (0,3)$ is the
mobility exponent parameterizing the nonlinear slip condition. The
partial-wetting regime implies the boundary condition $\partial_y h =
\mathrm{const.} > 0$ at the triple junction $\partial\{h > 0\}$ (nonzero
microscopic contact angle). Existence and uniqueness of traveling-wave
solutions to this problem under the constraint $\partial_y^2 h \to 0$ as $h \to
\infty$ have been proved in previous work by Chiricotto and Giacomelli in
[Commun. Appl. Ind. Math., 2(2):e-388, 16, 2011]. We are interested in the
asymptotics as $h \downarrow 0$ and $h \to \infty$. By reformulating the
problem as $h \downarrow 0$ 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 $h\downarrow0$ depending on
$n$.
  Together with the asymptotics as $h\to\infty$ characterizing the Cox-Voinov
law for the velocity-dependent macroscopic contact angle as found by
Giacomelli, the first author of this work, and Otto in [Nonlinearity,
29(9):2497-2536, 2016], 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.

Analysis of PDEs;Classical Analysis and ODEs;Dynamical Systems;Fluid Dynamics