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 = -∇·(|u|ⁿ∇Δu)- Δ(|u|^p-1u), 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 = p₀ = n + 1 +1 + 2/N for n ∈(0, 3/2), 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 self-similar solutions. For the Cauchy problem in R ^N × 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 = -Δ²u - Δ(|u|^p-1u) in R^N × 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.