Unstable sixth-order thin film equation: I. Blow-up similarity solutions

We study blow-up behaviour of solutions of the sixth-order thin film equation containing an unstable (backward parabolic) second-order term. By a formal matched expansion technique, we show that, for the first critical exponent where N is the space dimension, the free-boundary problem (FBP) with zero-height, zero-contact-angle, zero-moment, and zero-flux conditions at the interface admits a countable set of continuous branches of radially symmetric self-similar blow-up solutions where T > 0 is the blow-up time. We also study the Cauchy problem (CP) in R^N × R₊ and show that the corresponding self-similar family {u_k(x, t)} is countable and consists of solutions of maximal regularity, which are oscillatory near the interfaces. Actually, we show that compactly supported oscillatory blow-up profiles for the CP exist for all n ∈ (0, n_h), where is a heteroclinic bifurcation point for the ordinary differential equation involved. The FBP ceases to exist before, at n = 4/3 = 1.333....