On a parabolic equation in microelectromechanical systems with an external pressure

Abstract

The parabolic problem $u_t-\Delta u=\frac{\lambda f\left(x\right)}{{(1-u)}^2}+P$ 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 solutions to this equation. We first show that under certain initial conditions, there exist critical constants $P^{\ast }$ and ${\lambda }_P^{\ast }$ such that when $0\le P\le P^{\ast },0<\lambda \le {\lambda }_P^{\ast }$, there exists a global solution, while for $0\le P\le P^{\ast },\lambda >{\lambda }_P^{\ast }$ or $P>P^{\ast }$, the solution quenches in finite time. The estimates of voltage ${\lambda }_P^{\ast }$, quenching time $T$, and pressure term $P^{\ast }$ are investigated. The quenching set $\Sigma$ is proved to be a compact subset of $\Omega$ with an additional condition on $f\left(x\right)$, 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-sided bound estimate for the quenching solution but also obtain its asymptotic behavior near the quenching time.