Source-type solutions of the fourth-order unstable thin film equation

We consider the fourth-order thin film equation (TFE) u_t = -∇ · (|u|ⁿ∇Δu) - Δ(|u| ^p-1-u), n > 0, p > 1, with the unstable second-order diffusion term. We show that, for the first critical exponent p = p₀ = n+1+2/N for n ε s(0, 3/2), where N ≥ 1 is the space dimension, the free-boundary problem the with zero contact angle and zero-flux conditions admits continuous sets (branches) of self-similar similarity solutions of the form u(x, t) = t-^N/4+nN f(y), y = x/t1/t^4+nN. For the Cauchy problem, we describe families of self-similar patterns, which admit a regular limit as n → 0^- and converge to the similarity solutions of the semilinear unstable limit Cahn-Hilliard equation u_t = -Δ²u- Δ(|u|^p-1-u) studied earlier in [12]. Using both analytic and numerical evidence, we show that such solutions of the TFE are oscillatory and of changing sign near interfaces for all n ε (0, n_h), where the value n_h = 1.75865... characterizes a heteroclinic bifurcation of periodic solutions in a certain rescaled ODE. We also discuss the cases p ≠ p₀, the interface equation, and regular analytic approximations for such TFEs as an approach to the Cauchy problem.