On the steady states of thin film equations

This dissertation studies the steady state of thin film type equations. Different considerations of physical forces give different formulations of differential equations. We start with generalized thin film evolution and derive the second order elliptic equation for steady states. For the thin film driven by both van der Waals force and Born repulsion force, we define associated energy and obtain a classical energy minimizing problem by taking semi-limit. The solution has been proven to converge to a Dirac measure in the limit that repulsive force term tends to $0$. Asymptotic analysis show that the location of the spike would be a point on the boundary with maximal curvature. Furthermore, we neglect the Born repulsion force and study radial steady state solution for van der Waals force driven thin film equation. We link the volume constraint problem with a initial value ordinary different equation and analyze how radial steady state solution and associated energy depend on the average thickness