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

Abstract

In the current work we study a stochastic parabolic problem. The underlying problem is actually motivated by the study of an idealized electrically actuated MEMS (Micro-Electro-Mechanical System) device in the case of random fluctuations of the potential difference controlling the device. We first present the mathematical model and then we deduce some local existence results. Next for some particular versions of the model, regarding its boundary conditions, we derive quenching results as well as estimations of the probability for such singularity to occur. Additional numerical study of the problem in one dimension follows, investigating the problem further with respect to its quenching behaviour.

Existence and uniqueness for a coupled parabolic‐hyperbolic model of MEMS

“…The touchdown of a highly damped membrane corresponds to the quenching of a solution to the parabolic equation, and the quenching profile for a semilinear parabolic equation has been studied extensively in Guo [20]. Other quenching solutions for parabolic equations can be found in previous works [21,22], and the literature [23][24][25] of Kavallaris et al To be specific, Kavallaris [25] studies a similar case to (1.8), in which the right-hand side −𝛽 F 𝑓 (x)∕w 2 of (1.8) is replaced by −𝛽 F |x| 𝛽 ∕w p with specific 𝛽 and p, the resulting equation has a global-in-time solution for initial data close to 1, while quenching occurs for large 𝛽 F or small initial values. Similar conclusions also hold for a non-local version of (1.8).…”

“…More details can be found in Guo and Kavallaris [24], and Guo et al [26]. The recent publication of Kavallaris et al [23] generalizes the quenching solution to a stochastic parabolic equation modeling MEMS devices with random fluctuations of the potential difference.…”