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, u t − Δ u = λ g ( u ) ( 1 + δ | ∇ u | 2 ) , in a bounded domain Ω of R N . Here λ > 0 is a parameter related to the applied voltage, δ > 0 , g is a positive nondecreasing convex function, diverging as u → 1 . Firstly, we show that the solvability of the stationary problem − Δ w = λ g ( w ) ( 1 + δ | ∇ w | 2 ) with Dirichlet boundary condition is characterized by a parameter λ δ ∗ . Meanwhile it is shown that for λ > λ δ ∗ , any solution to the parabolic equation will quench ( u → 1 ) at a finite time. Secondly, we focus on estimating the quenching time T δ ∗ in terms of λ , λ δ ∗ , i.e. the quenching time T δ ∗ = O ( ( λ − λ δ ∗ ) − 1 2 ) , as λ → ( λ δ ∗ ) + .