Touchdown and related problems in electrostatic MEMS device equation

Abstract

Abstract.We study the electrostatic MEMS-device equation, ut − ∆u = λ|x| β (1−u) p , with Dirichlet boundary condition. First, we describe the touchdown of non-stationary solution in accordance with the total set of stationary solutions. Then, we classify radially symmetric stationary solutions and their radial Morse indices. Finally, we show the Morse-Smale property for radially symmetric non-stationary solutions. Mathematics Subject Classification (2000). Primary 35K55, 35J60; Secondary 74H35, 74G55, 74K15.

The existence and dynamic properties of a parabolic nonlocal MEMS equation

“…Interested readers can read the papers [2,5,6] for various results on the above equation. In [6] Lin and Yang by using a variational argument derived the following nonlocal MEMS equation:…”

“…The challenge is to build and understand mathematical models and the mechanisms for the various MEMS devices. Recently there has been a lot of study of the equations arising from MEMS by Esposito et al [1][2][3][4], Kavallaris et al [5], Lin and Yang [6], Ma and Wei [7], Flores et al [8][9][10] etc. Interested readers can read the book ''Modeling MEMS and NEMS'' [11], by Pelesko and Bernstein for the mathematical modeling and various applications of MEMS devices.…”

Variational approach to MEMS model with fringing field

“…When λ > λ * 0 , quenching (touchdown) phenomena occur at a finite time T * < ∞, that is, there exist x 0 ∈Ω and a sequence t n T * such that lim t n →T * u(x 0 , t n ) = 1. (See [8][9][10]15]. )…”

Global existence and quenching for a damped hyperbolic MEMS equation with the fringing field

“…For an elliptic and parabolic operator A, thanks to the maximum principle, we have the results of the time-global existence [12,19,23,25] for sufficiently small λ > 0, the quenching [12,18,23,25] for sufficiently large λ > 0, the connecting orbit [23], the Morse-Smale property [23], the location of the quenching point [17] and its stationary solution [5,6,7,11,13,23]. Also in the hyperbolic problem, we have similar results to those in the parabolic case, i.e., the global existence [3,26,38], the quenching [3,26,32,38], the estimate of the quenching time [32] and the singularity of the derivative [2].…”