On continuous branches of very singular similarity solutions of a stable thin film equation. I - The Cauchy problem

We consider 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. 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, the Cauchy problem is shown to admit countable continuous branches of source-type self-similar very singular solutions of the form u(x, t) = t(-N/4+nN) f(y), y = x/t(1/4+nN). 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 not equal p(0), 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 {p(l), l >= 0}. At p = p(l), 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 u(t) = -del center dot(vertical bar u vertical bar(n)del Delta u) in R-N x R+. 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 u(t) = -Delta(2)u. Numerics are used, wherever appropriate, to support the analysis.