### Abstract

Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of microelectromechanical systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a phenomenon known as quenching. For certain parameter regimes it is shown numerically that the singularity will form at multiple isolated points in the 1D strip case and along a ring of points in the radially symmetric two-dimensional case. The location of these touchdown points is accurately predicted by means of asymptotic expansions. The solution itself is shown to converge to a stable self-similar profile at the singularity point. Analytical calculations are verified by use of adaptive numerical methods which take advantage of symmetries exhibited by the underlying PDE to accurately resolve solutions very close to the singularity.

Original language | English (US) |
---|---|

Pages (from-to) | 935-958 |

Number of pages | 24 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 72 |

Issue number | 3 |

DOIs | |

State | Published - 2012 |

### Fingerprint

### Keywords

- Biharmonic equations
- Self-similar solutions
- Singularity formation
- Touchdown

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

**Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a mems capacitor.** / Lindsay, A. E.; Lega, Joceline C.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 72, no. 3, pp. 935-958. https://doi.org/10.1137/110832550

}

TY - JOUR

T1 - Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a mems capacitor

AU - Lindsay, A. E.

AU - Lega, Joceline C

PY - 2012

Y1 - 2012

N2 - Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of microelectromechanical systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a phenomenon known as quenching. For certain parameter regimes it is shown numerically that the singularity will form at multiple isolated points in the 1D strip case and along a ring of points in the radially symmetric two-dimensional case. The location of these touchdown points is accurately predicted by means of asymptotic expansions. The solution itself is shown to converge to a stable self-similar profile at the singularity point. Analytical calculations are verified by use of adaptive numerical methods which take advantage of symmetries exhibited by the underlying PDE to accurately resolve solutions very close to the singularity.

AB - Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of microelectromechanical systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a phenomenon known as quenching. For certain parameter regimes it is shown numerically that the singularity will form at multiple isolated points in the 1D strip case and along a ring of points in the radially symmetric two-dimensional case. The location of these touchdown points is accurately predicted by means of asymptotic expansions. The solution itself is shown to converge to a stable self-similar profile at the singularity point. Analytical calculations are verified by use of adaptive numerical methods which take advantage of symmetries exhibited by the underlying PDE to accurately resolve solutions very close to the singularity.

KW - Biharmonic equations

KW - Self-similar solutions

KW - Singularity formation

KW - Touchdown

UR - http://www.scopus.com/inward/record.url?scp=84865693350&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84865693350&partnerID=8YFLogxK

U2 - 10.1137/110832550

DO - 10.1137/110832550

M3 - Article

AN - SCOPUS:84865693350

VL - 72

SP - 935

EP - 958

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 3

ER -