Well-posedness of a fourth order evolution equation Modeling MEMS

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 $n\leq7$.