This thesis considers the thin-film equation ∂_tu = −∂_x (|u|ⁿ∂³x u) with respect to time t ≥ 0 and one dimensional space x ∈ R where n > 0. A special case of the thin-film equation is when the initial condition is u₀(x,0) = cδ(x). A solution with this initial condition is called a source type solution. A source type solution describes how a viscous droplet spreads over a solid flat surface with volume c > 0. Source type solutions are expected to have a self-similar form with u(x,t) = t^{-α}f (μ), μ = xt^{-α} and α = 1/(n+4) which reduces the equation into an ordinary boundary-value problem (|f (μ)|ⁿ f ′′′(μ))′ = α(μf (μ))′ with μf (μ)→0 as μ→±∞ and
∫_{-∞}^{∞} f (μ)dμ = c. A solution of this boundary-value problem is called a self-similar solution. This thesis presents a detailed discussion of the paper in 1992 by Bernis, Peletier & Williams on the existence and uniqueness of self-similar solutions to the thin-film equation together with its qualitative properties. Here, existence will be proven by using a shooting method. Additionally, the thin-film equation will be derived from the Navier-Stokes equations using a lubrication approximation. Furthermore, a numerical construction of the
self-similar solution is presented to visualize its behavior. The results demonstrate that the solution exhibits key qualitative features such as compact support and conservation of mass. ... Read more