This thesis treats the thin-film equation which models the film height $h$ for a viscous film in the complete wetting regime. We show existence and uniqueness to the thin-film equation with mobility m(h) = hⁿ and mobility exponent n∈ (1,3/2)∪ (3/2,3). The thin-film equation is rewritten as an abstract Cauchy problem and usage of semi-group theory yields maximal L^p-regularity for the linearized problem. With a fixed point argument, analogous to the one used by Giacomelli, Gnann, Knüpfer and Otto in their 2014 paper, the nonlinear problem is treated. Under a smallness condition on the initial value to a suitably transformed version of the thin-film equation, we obtain a solution in L^p(0,∞;H_{k-2,α-1/2})∩ \dot{W}^1,p(0,∞;H_{k+2,α+1\2}), where the H-spaces denote weighted Sobolev spaces. The novelty of this work lies in the usage of L^p-spaces in time, where the existing literature only deals with L²-spaces. It is found that the L^p setting allows for treatment of all n∈ (1,3/2)∪ (3/2,3).

This thesis considers the thin-film equation in partial wetting. The mobility in this equation is given by h³+λ^3-nhⁿ, where h is the film height, λ is the slip length and n is the mobility exponent. The partial wetting regime implies the boundary condition dh/dz>0 at the triple junction. The asymptotics as h↓0 are investigated. This is done by using a dynamical system for the error between the solution and the microscopic contact angle. Using the linearized version of the dynamical system, values for n when resonances occur are found. These resonances lead to a different behaviour for the solution as h↓0, so the asymptotics are found to be different for different values of n. Together with the asymptotics for h→∞ as found in [Giacomelli et al., 2016], the solution to the thin-film equation in partial wetting can be characterized. Also, via this solution, the relation between the microscopic and macroscopic contact angles can be analyzed. From the main result of this thesis, it can be seen that the macroscopic Tanner law for the contact angle depends smoothly on the microscopic contact angle.