Unstable sixth-order thin film equation: II. Global similarity patterns

We continue the study begun in part I (Evans J D, Galaktionov V A and King J R 2007 Unstable sixth-order thin film equation: I. Blow-up of similarity solutions Nonlinearity 20 1799-841) of asymptotic large time behaviour of global solutions of the sixth-order thin film equation with bounded integrable initial data. We show that for the first critical exponent, where N is the space dimension, the free-boundary problem with zero-contact-angle, zero-moment and zero-flux conditions at the interface admits continuous families (branches) of radially symmetric self-similar solutions defined for all t > 0, We also study the Cauchy problem, for which we construct global similarity solutions of the maximal regularity, these being oscillatory near the interfaces for n ∈ (0, n_h), where is a 'heteroclinic bifurcation' point for a related nonlinear ordinary differential equation. We use various concepts based on the branching of sufficiently small solutions from the known eigenfunctions of the linear rescaled operator corresponding to n ≤ 0.