On the quenching behavior of the MEMS with fringing field

Abstract

The singular parabolic problem $u_t-\triangle u=\lambda{\frac{1+\delta|\nabla u|^2}{(1-u)^2}}$ on a bounded domain $\Omega$ of $\mathbb{R}^n$ with Dirichlet boundary condition, models the Microelectromechanical systems (MEMS) device with fringing field. In this paper, we focus on the quenching behavior of the solution to this equation. We first show that there exists a critical value $\lambda_\delta^*>0$ such that if $0<\lambda<\lambda_\delta^*$, all solutions exist globally; while for $\lambda>\lambda_\delta^*$, all the solution will quench in finite time. The estimate of the quenching time in terms of large voltage $\lambda$ is investigated. Furthermore, the quenching set is a compact subset of $\Omega$, provided $\Omega$ is a convex bounded domain in $\mathbb{R}^n$. In particular, if the domain $\Omega$ is radially symmetric, then the origin is the only quenching point. We not only derive the one-side estimate of the quenching rate, but also further study the refined asymptotic behavior of the finite quenching solution.