The structure of stationary solutions to a micro-electro mechanical system with fringing field

Guo, Jong‐Shenq; Huang, Bo-Chih; Wakasa, Tohru; Wang, Chi-Jen; Yu, Cherng-Yih

doi:10.1016/j.jde.2020.05.041

Cited by 5 publications

(4 citation statements)

References 22 publications

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“…κ = −1) is obtained. Therefore, these trajectories correspond to the radially symmetric stationary solution in (8). Moreover, these trajectories exist in {a > 0}, so U(r) < 1 (r ∈ (0, +∞)) from the transformation in case 1.…”

Section: Proof Of Theoremmentioning

confidence: 96%

“…This paper is devoted to the study of the radially symmetric solutions of (7): 2 1 − U = 0, = d dr and = d 2 dr 2 (8) with U = U(r) and r = |x| > 0. In elliptic equations, it is important to consider radially symmetric solutions when we investigate the structure of these solutions.…”

Section: Preliminariesmentioning

confidence: 99%

“…Therefore, there are no trajectories connecting equilibria in Φ 4 \{a < 0} including equilibria at infinity. Although each orbit in this region may imply the existence of solutions to (8), this analysis method does not investigate detailed information on the shape and asymptotic behaviour of the solutions. We leave it open here.…”

Section: Remark 11mentioning

confidence: 99%

See 2 more Smart Citations

Radially symmetric stationary solutions for a MEMS type reaction–diffusion equation with fringing field

Ichida

Sakamoto

2022

Nonlinearity

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Radially symmetric stationary solutions for a micro-electro-mechanical system (MEMS) type reaction–diffusion equation with fringing field are considered. This equation arises in the study of the MEMS devices. This paper is devoted to the study of the existence of these solutions, information about their shape, and their asymptotic behaviour. These are studied by applying the framework that combines Poincaré type compactification, classical dynamical systems theory, and geometric methods for desingularization of vector fields called the blow-up technique.

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Section: Proof Of Theoremmentioning

confidence: 96%

Section: Preliminariesmentioning

confidence: 99%

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Radially symmetric stationary solutions for a MEMS type reaction–diffusion equation with fringing field

Ichida

Sakamoto

2022

Nonlinearity

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“…Qualitative properties of these type equations have been studied in [7, 8, 17-20, 24, 26] and the references therein. When f is replaced by 1 + |∇u| 2 , the MEMS equations with fringing fields are discussed in [5,14].…”

Section: Introductionmentioning

confidence: 99%

On semilinear elliptic equation with negative exponent arising from a closed MEMS model

Chen¹,

Wang²,

Zhu³

2022

Preprint

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This paper is concerned with the elliptic equation −∆u = λ (a−u) p in a connected, bounded C 2 domain Ω of R N subject to zero Dirichlet boundary conditions, where λ > 0, N ≥ 1, p > 0 and a : Ω → [0, 1] vanishes at the boundary with the rate dist(x, ∂Ω) γ for γ > 0.When p = 2 and N = 2, this equation models the closed Micro-Electromechanical Systems devices, where the elastic membrane sticks the curved ground plate on the boundary, but insulating on the boundary. The function a shapes the curved ground plate.Our aim in this paper is to study qualitative properties of minimal solutions of this equation when λ > 0, p > 0 and to show how the boundary decaying of a works on the minimal solutions and the pull-in voltage. Particularly, we give a complete analysis for the stability of the minimal solutions.

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On a parabolic equation in microelectromechanical systems with an external pressure

Zhang,

Wang

2024

Math Methods in App Sciences

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The parabolic problem on a bounded domain of with Dirichlet boundary condition models the microelectromechanical systems (MEMS) device with an external pressure term. In this paper, we classify the behavior of the solutions to this equation. We first show that under certain initial conditions, there exist critical constants and such that when , there exists a global solution, while for or , the solution quenches in finite time. The estimates of voltage , quenching time , and pressure term are investigated. The quenching set is proved to be a compact subset of with an additional condition on , provided is a convex bounded set. In particular, if is radially symmetric, then the origin is the only quenching point. Furthermore, we not only derive the two‐sided bound estimate for the quenching solution but also obtain its asymptotic behavior near the quenching time.

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The structure of stationary solutions to a micro-electro mechanical system with fringing field

Cited by 5 publications

References 22 publications

Radially symmetric stationary solutions for a MEMS type reaction–diffusion equation with fringing field

Radially symmetric stationary solutions for a MEMS type reaction–diffusion equation with fringing field

On semilinear elliptic equation with negative exponent arising from a closed MEMS model

On a parabolic equation in microelectromechanical systems with an external pressure

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