## Abstract

In this work we study a non-local hyperbolic equation of the form

u(tt) = u(xx) + lambda/(1 - u)(2) (1 + alpha integral(1)(0) (1/(1 - u)) dx)(2),

with homogeneous Dirichlet boundary conditions and appropriate initial conditions. The problem models an idealized electrostatically actuated MEMS (Micro-Electro-Mechanical System) device. Initially we present the derivation of the model. Then we prove local existence of solutions for lambda > 0 and global existence for 0 < lambda < lambda_* for some positive lambda_*, with zero initial conditions; similar results are obtained for other initial data. For larger values of the parameter lambda, i.e., when lambda > lambda(+)* for some constant lambda(+)* >= lambda_* and with zero initial conditions, it is proved that the solution of the problem quenches in finite time; again similar results are obtained for other initial data. Finally the problem is solved numerically with a finite difference scheme. Various simulations of the solution of the problem are presented, illustrating the relevant theoretical results.

Original language | English |
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Pages (from-to) | 505-534 |

Number of pages | 30 |

Journal | Rocky Mountain Journal of Mathematics |

Volume | 41 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2011 |

## Keywords

- Electrostatic MEMS
- quenching of solution
- hyperbolic non-local problems
- ELECTROSTATIC MEMS
- QUENCHING PROBLEM
- EQUATIONS