### 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.

Language | English |
---|---|

Pages | 3913-3938 |

Number of pages | 26 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 38 |

Issue number | 8 |

DOIs | |

Status | Published - 1 Aug 2018 |

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### 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

### Cite this

*Discrete and Continuous Dynamical Systems- Series A*,

*38*(8), 3913-3938. https://doi.org/10.3934/dcds.2018170

**Gradient blow-up for a fourth-order quasilinear boussinesq-type equation.** / Álvarez-Caudevilla, Pablo; Evans, Jonathan D.; Galaktionov, Victor A.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems- Series A*, vol. 38, no. 8, pp. 3913-3938. https://doi.org/10.3934/dcds.2018170

}

TY - JOUR

T1 - Gradient blow-up for a fourth-order quasilinear boussinesq-type equation

AU - Álvarez-Caudevilla, Pablo

AU - Evans, Jonathan D.

AU - Galaktionov, Victor A.

PY - 2018/8/1

Y1 - 2018/8/1

N2 - The Cauchy problem for a fourth-order Boussinesq-type quasilinear wave equation (QWE–4) of the form utt = −(|u|nu)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.

AB - The Cauchy problem for a fourth-order Boussinesq-type quasilinear wave equation (QWE–4) of the form utt = −(|u|nu)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.

KW - Fourth-order quasilinear wave equation

KW - Gradient blow-up

KW - Self-similarity of the second kind

UR - http://www.scopus.com/inward/record.url?scp=85048033204&partnerID=8YFLogxK

U2 - 10.3934/dcds.2018170

DO - 10.3934/dcds.2018170

M3 - Article

VL - 38

SP - 3913

EP - 3938

JO - Discrete and Continuous Dynamical Systems

T2 - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 8

ER -