## Abstract

Micro-Electro Mechanical Systems (MEMS) are defined as very small structures that combine electrical and mechanical components on a common substrate. Here, the electrostatic-elastic case is considered, where an elastic membrane is allowed to deflect above a ground plate under the action of an electric potential, whose strength is proportional to a parameter λ. Such devices are commonly described by a parabolic partial differential equation that contains a singular nonlinear source term. The singularity in that term corresponds to the so-called "touchdown" phenomenon, where the membrane establishes contact with the ground plate. Touchdown is known to imply the non-existence of steady-state solutions and blow-up of solutions in finite time.

We study a recently proposed extension of that canonical model, where such singularities are avoided due to the introduction of a regularizing term involving a small "regularization" parameter ε. Methods from dynamical systems and geometric singular perturbation theory, in particular the desingularization technique known as "blow-up", allow for a precise description of steady-state solutions of the regularized model, as well as for a detailed resolution of the resulting bifurcation diagram. The interplay between the two principal model parameters ε and λ is emphasized; in particular, the focus is on the singular limit as both parameters tend to zero.

We study a recently proposed extension of that canonical model, where such singularities are avoided due to the introduction of a regularizing term involving a small "regularization" parameter ε. Methods from dynamical systems and geometric singular perturbation theory, in particular the desingularization technique known as "blow-up", allow for a precise description of steady-state solutions of the regularized model, as well as for a detailed resolution of the resulting bifurcation diagram. The interplay between the two principal model parameters ε and λ is emphasized; in particular, the focus is on the singular limit as both parameters tend to zero.

Original language | English |
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Pages (from-to) | 661-708 |

Journal | Siam Journal on Applied Dynamical Systems |

Volume | 18 |

Issue number | 2 |

Early online date | 11 Apr 2019 |

DOIs | |

Publication status | Published - 30 Jun 2019 |