On the partial differential equations of electrostatic MEMS devices II: Dynamic case

Ghoussoub, Nassif; Guo, Yujin

doi:10.1007/s00030-007-6004-1

Cited by 72 publications

(90 citation statements)

References 19 publications

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“…whenever β > 0, C r bends infinitely many times with respect to λ around λ * , while at most a finite number of bendings occur to C r when n ∈ [7,9]…”

Section: If There Ismentioning

confidence: 99%

“…First, the above profile of C r is observed numerically [8]. Next, some estimates on λ are given for nonradially symmetric case [8,9]. Finally, existence of the non-minimal non-radially symmetric stationary solution is proven by the variational method [4].…”

Section: If There Ismentioning

confidence: 99%

“…The original proof for p = 2 to the first case of this theorem is quite technical [9], it provides with some bounds of the blow-up time though. Here we give a simpler proof by applying Kaplan's method adopted by [2].…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…Here we give a simpler proof by applying Kaplan's method adopted by [2]. The proof of the second case is obtained by using classical techniques like maximum-principle or dynamical-systems aguments, [7,9], and we omit it. The final case is obtained by the method of [15].…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…The global-in-time behaviour of the non-stationary solution has been studied also for p = 2 ( [9]). More precisely, for λ < λ and λ > λ, there is a solution to (1.1) converging to u λ uniformly and any solution must touchdown in finite time, i.e.,…”

Section: Introductionmentioning

confidence: 99%

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Touchdown and related problems in electrostatic MEMS device equation

Kavallaris

Miyasita

Suzuki

2008

Nonlinear differ. equ. appl.

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Abstract.We study the electrostatic MEMS-device equation, ut − ∆u = λ|x| β (1−u) p , with Dirichlet boundary condition. First, we describe the touchdown of non-stationary solution in accordance with the total set of stationary solutions. Then, we classify radially symmetric stationary solutions and their radial Morse indices. Finally, we show the Morse-Smale property for radially symmetric non-stationary solutions. Mathematics Subject Classification (2000). Primary 35K55, 35J60; Secondary 74H35, 74G55, 74K15.

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“…whenever β > 0, C r bends infinitely many times with respect to λ around λ * , while at most a finite number of bendings occur to C r when n ∈ [7,9]…”

Section: If There Ismentioning

confidence: 99%

Section: If There Ismentioning

confidence: 99%

Section: Proof Of Theorem 11mentioning

confidence: 99%

Section: Proof Of Theorem 11mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Touchdown and related problems in electrostatic MEMS device equation

Kavallaris

Miyasita

Suzuki

2008

Nonlinear differ. equ. appl.

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Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity

Esposito

Ghoussoub

Guo

2007

Comm Pure Appl Math

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We study the branch of semistable and unstable solutions (i.e., those whose Morse index is at most 1) of the Dirichlet boundary value problem $-\Delta u = λf (x)/(1 − u)^2$ on a bounded domain $\Omega ⊂ R^N$ , which models—among other things—a simple electrostatic microelectromechanical system (MEMS) device. We extend the results of [11] relating to the minimal branch, by obtaining compactness along unstable branches for 1 ≤ N ≤ 7 on any domain \Omega and for a large class of “permittivity profiles” f . We also show the remarkable fact that powerlike profiles f (x) ≃ |x|^α can push back the critical dimension N = 7 of this problem by establishing compactness for the semistable branch on the unit ball, also for N≥8 and as long as $α>α_N =3N−14−4√6/4+2√6$. As a byproduct, we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for λ in a natural range. In all these results, the conditions on the space dimension and on the power of the profile are essentially sharp

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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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On the partial differential equations of electrostatic MEMS devices II: Dynamic case

Abstract: Abstract. This paper is a continuation of [9], where we analyzed steadystates of the nonlinear parabolic problem ut = ∆u − λf (x)

Cited by 72 publications

References 19 publications

Touchdown and related problems in electrostatic MEMS device equation

Touchdown and related problems in electrostatic MEMS device equation

Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity

On a parabolic equation in microelectromechanical systems with an external pressure

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