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

Abstract

In the current work, we study a nonlocal parabolic problem with Robin boundary conditions. The problem arises from the study of an idealized electrically actuated MEMS (micro‐electro‐mechanical system) device, when the ends of the device are attached or pinned to a cantilever. Initially, the steady‐state problem is investigated estimates of the pull‐in voltage are derived. In particular, a Pohožaev's type identity is also obtained, which then facilitates the derivation of an estimate of the pull‐in voltage for radially symmetric N‐dimensional domains. Next a detailed study of the time‐dependent problem is delivered and global‐in‐time as well as quenching results are obtained for generic and radially symmetric domains. The current work closes with a numerical investigation of the presented nonlocal model via an adaptive numerical method. Various numerical experiments are presented, verifying the previously derived analytical results as well as providing new insights on the qualitative behavior of the studied nonlocal model.

Impacts of noise on quenching of some models arising in MEMS technology

“…Also, for the case of a model with a general diffusion term σ (u) the interested reader can check [27]. When the two edges of the membrane are attached to a pair of torsional and translational springs, modelling a flexible nonideal support [12,54], see also Figure 2, then homogeneous boundary conditions of the form (1.1b), with β c = 0, are imposed together with the stochastic equation for the deformation u and complemented with initial condition (1.1c).…”

“…12) where σ 1 (z) := −κ(t)z clearly satisfies a Lispchitz condition for κ(t) bounded.10 O. Drosinou et al…”

“…For hyperbolic modifications of the deterministic variation of (1.1), an interested reader can check [17,23,30]. Finally, non-local alterations of parabolic and hyperbolic problems arising in MEMS technology are treated in [12,14,19,21,20,29,31,32,41,42,43]. Due to the presence of the term f (u) := 1 (1−u) 2 in (1.2a), the occurrence of a singular behaviour, called quenching, is observed when max x∈D u → 1.…”

A stochastic parabolic model of MEMS driven by fractional Brownian motion

“…Hyperbolic variations of ( 4)-( 6) have been investigated in Flores (2014), Guo (2010), Kavallaris et al (2015). Non-local parabolic and hyperbolic problems, arising in the modeling and control of MEMS devices, have been analyzed in Drosinou et al (2021), Duong and Zaag (2019), Guo et al (2009), Guo et al (2020), Guo and Kavallaris (2012), Kavallaris et al (2011), Kavallaris et al (2016), Kavallaris and Suzuki (2018), Miyasita (2017a), Miyasita (2017b), Miyasita (2015). Furthermore, a special case of (1)-( 3) for H = 1/2, i.e., when then dynamics are driven by Brownian motion, has been previously investigated in Drosinou et al (2022), Kavallaris (2018).…”

Diffusion-induced blowup solutions for the shadow limit model of a singular Gierer-Meinhardt system

“…Additionally, in [DZ19] a singular solution associated with a duality concept to blowup phenomenon, called quenching (or touch-down in MEMS literature), is constructed. In particular, the authors in [DZ19] developed further the idea of [MZ97a] to describe the quenching behaviour of a non-local problem arising from MEMS industry (see more in [DKN20], [GK12], [GS15], [KLN16] and the references therein).…”