## Abstract

The Cauchy problem for a fourth-order Boussinesq-type quasilinear wave equation (QWE–4) of the form utt = −(|u|^{n}u)xxxx in R × R_{+}, with a fixed exponent n > 0, and bounded smooth initial data, is considered. Self-similar single-point gradient blow-up solutions are studied. It is shown that such singular solutions exist and satisfy the case of the so-called self-similarity of the second type. Together with an essential and, often, key use of numerical methods to describe possible types of gradient blow-up, a “homotopy” approach is applied that traces out the behaviour of such singularity patterns as n → 0^{+}, when the classic linear beam equation occurs utt = −uxxxx, with simple, better-known and understandable evolution properties.

Original language | English |
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Pages (from-to) | 3913-3938 |

Number of pages | 26 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 38 |

Issue number | 8 |

DOIs | |

Publication status | Published - 1 Aug 2018 |

## Keywords

- Fourth-order quasilinear wave equation
- Gradient blow-up
- Self-similarity of the second kind

## ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics