On A Parabolic Equation in MEMS with An External Pressure

TL;DR

This paper analyzes a parabolic PDE modeling MEMS devices with external pressure, classifying solution behaviors, identifying critical parameters for quenching, and studying the asymptotic and spatial properties of solutions.

Contribution

It provides a comprehensive classification of solution behaviors, critical parameter estimates, and spatial quenching set characterization for the MEMS model with external pressure.

Findings

Existence of critical constants P* and λ_P* for global solutions and quenching.

Quenching occurs in finite time when parameters exceed critical values.

The quenching set is a compact subset of the domain, with specific symmetry properties.

Abstract

The parabolic problem $u_t-\Delta u=\frac{\lambda f(x)}{(1-u)^2}+P$ on a bounded domain $\Omega$ of $R^n$ with Dirichlet boundary condition models the microelectromechanical systems(MEMS) device with an external pressure term. In this paper, we classify the behavior of the solution to this equation. We first show that under certain initial conditions, there exists critical constants $P^*$ and $\lambda_P^*$ such that when $0\leq P\leq P^*$, $0<\lambda\leq \lambda_P^*$, there exists a global solution, while for $0\leq P\leq P^*,\lambda>\lambda_P^*$ or $P>P^*$, the solution quenches in finite time. The estimate of voltage $\lambda_P^*$, quenching time $T$ and pressure term $P^*$ are investigated. The quenching set $\Sigma$ is proved to be a compact subset of $\Omega$ with an additional condition, provided $\Omega\subset R^n$ is a convex bounded set. In particular, if $\Omega$ is radially symmetric, then the origin is the only quenching point. Furthermore, we not only derive the two-side bound estimate for the quenching solution, but also study the asymptotic behavior of the quenching solution in finite time.