Prof. T.P. Witelski (Duke University)

Dynamics for a critical-case unstable generalized thin film equation

We examine the dynamics of blow-up singularities in a critical-case unstable thin film equation. This is a nonlinear fourth-order degenerate parabolic PDE derived from a generalized model for the free-surface evolution of lubrication flows of thin viscous films. For a special balance between destabilizing second-order terms and regularizing fourth-order terms, this equation has a very rich set of dynamics including families of similarity solutions for finite-time blow-up and infinite-time spreading. The structure and stability of the steady-states and the compactly-supported similarity solutions is studied. This is joint work done with Andrew Bernoff and Andrea Bertozzi.