Well-posedness of a fourth order evolution equation Modeling MEMS

TL;DR

This paper proves the well-posedness of a fourth order evolution equation modeling MEMS in higher dimensions using Faedo-Galerkin methods, extending previous results limited to lower dimensions.

Contribution

It introduces a Faedo-Galerkin approach to establish well-posedness for the equation in dimensions up to 7, surpassing prior semigroup-based results limited to dimension 2.

Findings

Well-posedness established for dimensions n ≤ 7

Extension of methods from parabolic to hyperbolic equations

Improved understanding of MEMS-related PDE models

Abstract

We consider a fourth order evolution equation involving a singular nonlinear term $\frac{\lambda}{(1-u)^{2}}$ in a bounded domain $\Omega\subset\R^{n}$. This equation arises in the modeling of microelectromechanical systems. We first investigate the well-posedness of a fourth order parabolic equation which has been studied in \cite{Lau}, where the authors, by the semigroup argument, obtained the well-posedness of this equation for $n\leq2$. Instead of semigroup method, we use the Faedo-Galerkin technique to construct a unique solution of the fourth order parabolic equation for $n\leq7$, which improves and completes the result of \cite{Lau}. Besides, the well-posedness of the corresponding fourth order hyperbolic equation is obtained by the similar argument for $n\leq7$.