We discuss the fourth-order thin film equation with a stable second-order diffusion term, in the context of a standard free-boundary problem with zero height, zero contact angle and zero-flux conditions imposed at an interface. For the first critical exponent where N ≥ 1 is the space dimension, there are continuous sets (branches) of source-type very singular self-similar solutions of the form For p ≠ p0, the set of very singular self-similar solutions is shown to be finite and consists of a countable family of branches (in the parameter p) of similarity profiles that originate at a sequence of critical exponents {pl, l ≥ 0}. At p = pl, these branches appear via a non-linear bifurcation mechanism from a countable set of second kind similarity solutions of the pure thin film equation Such solutions are detected by a combination of linear and non-linear ‘Hermitian spectral theory’, which allows the application of an analytical n-branching approach. In order to connect with the Cauchy problem in Part I, we identify the cauchy problem solutions as suitable ‘limits’ of the free-boundary problem solutions.

We consider the fourth-order thin film equation, with a stable second-order diffusion term. For the first critical exponent, where N ≥ 1 is the space dimension, the Cauchy problem is shown to admit countable continuous branches of source-type self-similar very singular solutions of the form These solutions are inherently oscillatory in nature and will be shown in Part II to be the limit of appropriate free-boundary problem solutions. For p ≠ p0, the set of very singular solutions is shown to be finite and to be consisting of a countable family of branches (in the parameter p) of similarity profiles that originate at a sequence of critical exponents {pl, l ≥ 0}. At p = pl, these branches appear via a non-linear bifurcation mechanism from a countable set of similarity solutions of the second kind of the pure thin film equation Such solutions are detected by the ‘Hermitian spectral theory’, which allows an analytical n-branching approach. As such, a continuous path as n → 0+ can be constructed from the eigenfunctions of the linear rescaled operator for n = 0, i.e. for the bi-harmonic equation ut = −Δ2u. Numerics are used, wherever appropriate, to support the analysis.