### Abstract

In this work we study the semilinear wave equation of the form

utt = uxx + lambda/(1 − u)2;

with homogeneous Dirichlet boundary conditions and suitable initial conditions, which, under appropriate circumstances, serves as a model of an idealized electrostatically actuated MEMS device. First we establish local existence of the solutions of the problem for any lambda > 0: Then we focus on the singular behaviour of the solution, which occurs through

finite-time quenching, i.e. when ||u(·; t)||1 → 1 as t →t*− < ∞, investigating both conditions for quenching and the quenching profile of u: To this end, the non-existence of a regular similarity solution near a quenching point is first shown and then a formal asymptotic expansion is used to determine the local form of the solution. Finally, using a finite difference scheme, we solve the problem numerically, illustrating the preceding results.

utt = uxx + lambda/(1 − u)2;

with homogeneous Dirichlet boundary conditions and suitable initial conditions, which, under appropriate circumstances, serves as a model of an idealized electrostatically actuated MEMS device. First we establish local existence of the solutions of the problem for any lambda > 0: Then we focus on the singular behaviour of the solution, which occurs through

finite-time quenching, i.e. when ||u(·; t)||1 → 1 as t →t*− < ∞, investigating both conditions for quenching and the quenching profile of u: To this end, the non-existence of a regular similarity solution near a quenching point is first shown and then a formal asymptotic expansion is used to determine the local form of the solution. Finally, using a finite difference scheme, we solve the problem numerically, illustrating the preceding results.

Original language | English |
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Article number | 13 |

Pages (from-to) | 1009-1037 |

Number of pages | 29 |

Journal | Discrete and Continuous Dynamical Systems-Series A |

Volume | 35 |

Issue number | 3 |

Early online date | 1 Oct 2014 |

DOIs | |

Publication status | Published - Mar 2015 |

### Keywords

- Electrostatic MEMS
- Quenching of Solution
- Hyperbolic Problems

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## Profiles

## Andrew Alfred Lacey

- School of Mathematical & Computer Sciences - Professor
- School of Mathematical & Computer Sciences, Mathematics - Professor

Person: Academic (Research & Teaching)

## Cite this

Lacey, A. A., Kavallaris, N., Nikolopoulos, C., & Tzanetis, D. (2015). On the Quenching Behaviour of a Semilinear Wave Equation Modelling MEMS Technology.

*Discrete and Continuous Dynamical Systems-Series A*,*35*(3), 1009-1037. [13]. https://doi.org/10.3934/dcds.2015.35.1009