## 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) | 358-399 |

Number of pages | 42 |

Journal | European Journal of Applied Mathematics |

Volume | 30 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2019 Apr 1 |

## Keywords

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

## ASJC Scopus subject areas

- Applied Mathematics