We consider the thin-film equation for a class of free boundary conditions modelling friction at the contact line, as introduced by E and Ren. Our analysis focuses on formal long-time asymptotics of solutions in the perfect wetting regime. In particular, through the analysis of quasi-self-similar solutions, we characterize the profile and the spreading rate of solutions depending on the strength of friction at the contact line, as well as their (global or local) corrections, which are due to the dynamical nature of the free boundary conditions. These results are complemented with full transient numerical solutions of the free boundary problem.

We are interested in traveling-wave solutions to the thin-film equation with zero microscopic contact angle (in the sense of complete wetting without precursor) and inhomogeneous mobility h³+λ^3?nhⁿ, where h, λ, and n ϵ (3/2, 7/3) denote film height, slip parameter, and mobility exponent, respectively. Existence and uniqueness of these solutions have been established by Maria Chiricotto and the first of the authors in previous work under the assumption of subquadratic growth as h → ∞. In the present work we investigate the asymptotics of solutions as h ↘ 0 (the contact-line region) and h → ∞. As h ↘ 0 we observe, to leading order, the same asymptotics as for traveling waves or source-type self-similar solutions to the thin-film equation with homogeneous mobility hn and we additionally characterize corrections to this law. Moreover, as h → ∞ we identify, to leading order, the logarithmic Tanner profile, i.e. the solution to the corresponding unperturbed problem with λ = 0 that determines the apparent macroscopic contact angle. Besides higher-order terms, corrections turn out to affect the asymptotic law as h → ∞ only by setting the length scale in the logarithmic Tanner profile. Moreover, we prove that both the correction and the length scale depend smoothly on n. Hence, in line with the common philosophy, the precise modeling of liquid-solid interactions (within our model, the mobility exponent) does not affect the qualitative macroscopic properties of the film.

We are interested in the thin-film equation with zero-contact angle and quadratic mobility, modeling the spreading of a thin liquid film, driven by capillarity and limited by viscosity in conjunction with a Navier-slip condition at the substrate. This degenerate fourth-order parabolic equation has the contact line as a free boundary. From the analysis of the self-similar source-type solution, one expects that the solution is smooth only as a function of two variables (x, x^β) (where x denotes the distance from the contact line) with β=13-14≈0.6514 irrational. Therefore, the well-posedness theory is more subtle than in case of linear mobility (coming from Darcy dynamics) or in case of the second-order counterpart (the porous medium equation).Here, we prove global existence and uniqueness for one-dimensional initial data that are close to traveling waves. The main ingredients are maximal regularity estimates in weighted L²-spaces for the linearized evolution, after suitable subtraction of a(t)+b(t)x^β-terms.

In one space dimension, we consider source-type (self-similar) solutions to the thin-film equation with vanishing slope at the edge of their support (zero contact-angle condition) in the range of mobility exponents n\in\left(\frac 3 2,3\right). This range contains the physically relevant case n=2 (Navier slip). The existence and (up to a spatial scaling) uniqueness of these solutions has been established in [3] (Bernis, F., Peletier, L. A. & Williams, S. M. (1992) Source type solutions of a fourth-order nonlinear degenerate parabolic equation. Nonlinear Anal. 18, 217-234). It is also shown there that the leading-order expansion near the edge of the support coincides with that of a travelling-wave solution. In this paper we substantially sharpen this result, proving that the higher order correction is analytic with respect to two variables: the first one is just the spatial variable whereas the second one is a (generically irrational, in particular for n=2) power of it, which naturally emerges from a linearisation of the operator around the travelling-wave solution. This result shows that - as opposed to the case of n=1 (Darcy) or to the case of the porous medium equation (the second-order analogue of the thin-film equation) - in this range of mobility exponents, source-type solutions are not smooth at the edge of their support even when the behaviour of the travelling wave is factored off. We expect the same singular behaviour for a generic solution to the thin-film equation near its moving contact line. As a consequence, we expect a (short-time or small-data) well-posedness theory - of which this paper is a natural prerequisite - to be more involved than in the case n=1.