Estimates for the quenching time of a MEMS equation with fringing field

In this paper we consider a zero initial–boundary value parabolic problem in MEMS with fringing field, ut−Δu=λg(u)(1+δ|∇u|2)ut−Δu=λg(u)(1+δ|∇u|2), in a bounded domain ΩΩ of RNRN. Here λ>0λ>0 is a parameter related to the applied voltage, δ>0δ>0, gg is a positive nondecreasing convex function, diverging as u→1u→1. Firstly, we show that the solvability of the stationary problem −Δw=λg(w)(1+δ|∇w|2)−Δw=λg(w)(1+δ|∇w|2) with Dirichlet boundary condition is characterized by a parameter λδ∗. Meanwhile it is shown that for λ>λδ∗, any solution to the parabolic equation will quench (u→1u→1) at a finite time. Secondly, we focus on estimating the quenching time Tδ∗ in terms of λλ, λδ∗, i.e. the quenching time Tδ∗=O((λ−λδ∗)−12), as λ→(λδ∗)+.