On the quenching of a nonlocal parabolic problem arising in electrostatic MEMS control

Abstract

Abstract. We consider a nonlocal parabolic model for a micro-electro-mechanical system. Specifically, for a radially symmetric problem with monotonic initial data, it is shown that the solution quenches, so that touchdown occurs in the device, in a situation where there is no steady state. It is also shown that quenching occurs at a single point and a bound on the approach to touchdown is obtained. Numerical simulations illustrating the results are given.

A study of a nonlocal problem with Robin boundary conditions arising from technology

“…The quenching behavior of the nonlocal Equation () associated with Dirichlet boundary ( ) has been treated in Kavallaris et al 20 and in references therein as well as in previous works 21–23 . Also, non‐local alterations of parabolic and hyperbolic problems arising in MEMS technology were tackled in previous works 5,7,20–22,24,25 .…”

“…The quenching behavior of the nonlocal Equation () associated with Dirichlet boundary ( ) has been treated in Kavallaris et al 20 and in references therein as well as in previous works 21–23 . Also, non‐local alterations of parabolic and hyperbolic problems arising in MEMS technology were tackled in previous works 5,7,20–22,24,25 . However, to the best of our knowledge, there are not similar studies available in the literature for the Robin problem (0 < β < + ∞ ,) so in the current work we study problem (1.1) and we extend some of the results given in Guo 1 for the local problem, but we also deliver a further investigation related to the steady‐state problem and the quenching behavior of the time‐dependent problem.…”

“…However, to the best of our knowledge, there are not similar studies available in the literature for the Robin problem (0 < β < + ∞ ,) so in the current work we study problem (1.1) and we extend some of the results given in Guo 1 for the local problem, but we also deliver a further investigation related to the steady‐state problem and the quenching behavior of the time‐dependent problem. Our mathematical analysis is inspired by ideas developed in previous works; 20,22 however, important modifications are necessary due to the Robin boundary conditions. In particular, a new Pohožaev's type identity for Robin boundary conditions is derived which is then used to derive lower estimates of the pull‐in voltage.…”

“…In particular, a new Pohožaev's type identity for Robin boundary conditions is derived which is then used to derive lower estimates of the pull‐in voltage. Moreover, a novel argument, see Theorem 3.15, is developed to derive an upper estimate of the quenching rate; note that such a reasoning is missing from the approach used in Kavallaris et al 20 Still, the derivation of a key estimate for the nonlocal term, analogous to the one derived in Kavallaris et al 20 , Lemma 3.3 for the Dirichlet problem, needs more work for Robin problem (1.1) and it is finally derived under some extra restriction, cf. Lemma 3.10.…”

Quenching solutions of a stochastic parabolic problem arising in electrostatic MEMS control

“…In case σ ( u ) ≡ 0, then problem ()–() is reduced to its deterministic counterpart Actually, ()–() and its non‐local variations have attracted the attention of many researchers , because this kind of equations can model the operation of some (idealized) electrostatic actuated micro‐electro‐mechanical systems (MEMS), which have a wide variety of applications.…”