Blow-up similarity solutions of the fourth-order unstable thin film equation

We study blow-up behaviour of solutions of the fourth-order thin film equation u(t), = -del center dot (vertical bar u vertical bar(n)del u) - del(vertical bar u vertical bar(p-1)u), n > 0, p > 1, which contains a backward (unstable) diffusion term. Our main goal is a detailed study of the case of the first critical exponent p = P-0 = n + 1 + 2/N for n epsilon (0, 2/3), where N >= 1 is the space dimension. We show that the free-boundary problem with zero contact angle and zero-flux conditions admits continuous sets (branches) of blow-up selfsimilar solutions. For the Cauchy problem in R-N N x R+, we detect compactly supported blowup patterns, which have infinitely many oscillations near interfaces and exhibit "maximal" regularity there. As a key principle, we use the fact that, for small positive n, such solutions are close to the similarity solutions of the semilinear unstable limit Cahn-Hilliard equation u(t) = -Delta(2)u - Delta(vertical bar u vertical bar(p-1)u) in R-N x R-+,R- which are better understood and have been studied earlier [19]. We also discuss some general aspects of formation of self-similar blow-up singularities for other values of p.