The abstract quasilinear Cauchy problem for a MEMS model with two free boundaries

TL;DR

This paper analyzes a mathematical model of a MEMS device with two elastic membranes, proving local well-posedness, conditions for global solutions, and convergence to steady states and narrow gap limits.

Contribution

It reformulates the MEMS model as a quasilinear evolution equation and establishes well-posedness, stability, and asymptotic behavior, extending previous elliptic-parabolic analyses.

Findings

Local well-posedness for any source voltage.

Global solutions for small voltages.

Finite-time blow-up for large voltages.

Abstract

In this paper, we reformulate a mathematical model for the dynamics of an idealized electrostatically actuated MEMS device with two elastic membranes as an initial value problem for an abstract quasilinear evolution equation. Applying the Contraction Mapping Theorem, it is shown that the model is locally well-posed in time for any value of the source voltage of the device. In addition it is proven that the MEMS model considered here possesses global solutions for small source voltages whereas for large source voltages solutions of the model have a finite maximal existence time. Furthermore, we comment on the relationship of our model to its stationary version and to its small aspect ratio limit by showing that there exists a unique exponentially stable steady state and by proving convergence towards a solution of the narrow gap model in the vanishing aspect ratio limit. Our results extend the discussion of the elliptic-parabolic MEMS model presented in [Martin Kohlmann: On an elliptic-parabolic MEMS model with two free boundaries, 2014] leading to a Cauchy problem for a semilinear abstract evolution equation.