## Abstract

The formation of finite time singularities in a nonlinear parabolic fourth order partial differential equation (PDE) is investigated for a variety of two-dimensional geometries. The PDE is a variant of a canonical model for Micro-Electro Mechanical systems (MEMS). The singularities are observed to form at specific points in the domain and correspond to solutions whose values remain finite but whose derivatives diverge as the finite time singularity is approached. This phenomenon is known as quenching. An asymptotic analysis reveals that the quenching set can be predicted by simple geometric considerations suggesting that the phenomenon described is generic to higher order parabolic equations which exhibit finite time singularity.

Original language | English (US) |
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Pages (from-to) | 807-834 |

Number of pages | 28 |

Journal | Journal of Nonlinear Science |

Volume | 23 |

Issue number | 5 |

DOIs | |

State | Published - Oct 2013 |

## Keywords

- Bi-Laplace equations
- Finite time singularity
- MEMS
- Nanotechnology
- Singular perturbation theory

## ASJC Scopus subject areas

- Modeling and Simulation
- Engineering(all)
- Applied Mathematics