On continuous branches of very singular similarity solutions of the stable thin film equation. II - Free-boundary problems

We discuss the fourth-order thin film equation u(t) = -del center dot (vertical bar u vertical bar(n del)Delta u) + Delta(vertical bar u vertical bar(p-1)u), where n > 0, p > 1, 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 p = p(0) = n + 1 + 2/N for n is an element of (0, 3/2), where N >= 1 is the space dimension, there are continuous sets (branches) of source-type very singular self-similar solutions of the form u(x, t) = t(-N/4+nN) f(y), y = x/t(1/4+nN). For p not equal p(0), 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 {p(l), l >= 0}. At p = p(l), these branches appear via a non-linear bifurcation mechanism from a countable set of second kind similarity solutions of the pure thin film equation u(t) = -del center dot(vertical bar u vertical bar(n)del Delta u) in R-N x R+. 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.