On a general family of nonautonomous elliptic and parabolic equations

We consider the parabolic equation u _t ? Δu = λ f (x)g(u) as well as the corresponding elliptic problem, with a nonnegative profile f and a positive nondecreasing convex function g verifying ${\lim_{u \to 1^-} g(u) = \infty}$ . Our study is motivated by a simplified Micro-Electromechanical Systems (MEMS) device model. We extend or improve many qualitative and quantitative results for the MEMS modeling to this very general setting, which help us to understand more about the influence of f on the pull-in voltage λ* and the quenching phenomenon. Especially, we show some new estimates for λ* and the quenching time T.