### Abstract

We investigate self-similar sign-changing solutions to the thin-film equation, h_{t} = −(|h|^{n}h_{xxx})_{x}, on the semi-infinite domain x ≥ 0 with zero-pressure-type boundary conditions h = h_{xx} = 0 imposed at the origin. In particular, we identify classes of first- and second-kind compactly supported self-similar solutions (with a free-boundary x = s(t) = Lt^{β}) and consider how these solutions depend on the mobility exponent n; multiple solutions can exist with the same number of sign changes. For n = 0, we also construct first-kind self-similar solutions on the entire half-line x ≥ 0 and show that they act as limiting cases for sequences of compactly supported solutions in the limit of infinitely many sign changes. In addition, at n = 1, we highlight accumulation point-like behaviour of sign-changes local to the moving interface x = s(t). We conclude with a numerical investigation of solutions to the full time-dependent partial differential equation (based on a non-local regularisation of the mobility coefficient) and discuss the computational results in relation to the self-similar solutions.

Original language | English |
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Pages (from-to) | 1-42 |

Number of pages | 42 |

Journal | European Journal of Applied Mathematics |

DOIs | |

Publication status | Accepted/In press - 2018 Apr 2 |

### Keywords

- Second-kind solutions
- Sign-changing solutions
- Similarity solutions
- Thin-film equation

### ASJC Scopus subject areas

- Applied Mathematics

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## Cite this

*European Journal of Applied Mathematics*, 1-42. https://doi.org/10.1017/S095679251800013X