A fourth-order model for MEMS with clamped boundary conditions

TL;DR

This paper analyzes a fourth-order PDE modeling MEMS devices with clamped boundary conditions, revealing existence thresholds for solutions, well-posedness, and singularity formation related to the voltage parameter.

Contribution

It introduces a fourth-order model for MEMS with boundary conditions, establishing solution existence thresholds and analyzing dynamic behaviors including singularities.

Findings

Existence threshold for stationary solutions at critical voltage $\\lambda_*$.

Well-posedness results for hyperbolic and parabolic problems.

Finite time singularities occur when voltage exceeds threshold.

Abstract

The dynamical and stationary behaviors of a fourth-order equation in the unit ball with clamped boundary conditions and a singular reaction term are investigated. The equation arises in the modeling of microelectromechanical systems (MEMS) and includes a positive voltage parameter $\lambda$. It is shown that there is a threshold value $\lambda_*>0$ of the voltage parameter such that no radially symmetric stationary solution exists for $\lambda>\lambda_*$, while at least two such solutions exist for $\lambda\in (0,\lambda_*)$. Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as the occurrence of finite time singularities when $\lambda>\lambda_*$.