## Abstract

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.

Original language | English |
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Pages (from-to) | 735-760 |

Number of pages | 26 |

Journal | European Journal of Applied Mathematics |

Volume | 24 |

Issue number | 5 |

DOIs | |

Publication status | Published - Oct 2013 |

Externally published | Yes |

## Keywords

- Degenerate parabolic equations
- Invariant manifolds
- Key words: Self-similar solutions
- Singular nonlinear boundary value problems
- Smoothness and regularity of solutions