On a nonlocal parabolic problem arising in electrostatic MEMS control

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

“…where we always have 0 ≤ w < 1 in Ω for a (classical) solution of (2.1). Let λ * := sup{λ > 0 : problem (2.1) admits a classical solution}, (2.2) then λ * < ∞ for any dimension N ≥ 1, see [8,9]. For more on the structure of the solution set of (2.1) see [9].…”

“…The quenching behaviour of the solutions of (1.1)-(1.3) has been also studied in [8,9,13] but questions regarding : (i) occurrence of quenching for λ > λ * where λ * is defined by (2.2); (ii) determination of the quenching rate; (iii) establishment of the form of the quenching set; were left open. These questions are addressed in the current work for radially symmetric problems with the extra assumption that the initial data decreases with distance from the centre of the domain.…”

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

“…Although we can not apply the comparison principle owing to the nonlocal term, in the elliptic and parabolic problem, there are similar results to those without nonlocal term, i.e., the global existence [14,16,20], its asymptotic behaviour [20], the quenching [14,16,20] and its stationary solution [14,16,20,22,33,36]. For a hyperbolic and damped hyperbolic operator A, we have the results of the global existence [15,22] and the quenching [22] for Ω = (0, 1).…”

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.…”