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.
We consider the formation of finite-time quenching singularities for solutions of semi-linear wave equations with negative power nonlinearities, as can model micro-electro-mechanical systems (MEMS). For radial initial data we obtain, formally, the existence of a sequence of quenching self-similar solutions. Also from formal asymptotic analysis, a solution to the PDE which is radially symmetric and increases strictly monotonically with distance from the origin quenches at the origin like an explicit spatially independent solution. The latter analysis and numerical experiments suggest a detailed…
No touchdown at zero points of the permittivity profile for the MEMS problem
研究了微机电系统（MEMS）模型中半线性热方程的猝灭行为。这个问题涉及到一个源项与空间相关的电位，给定的介电常数的配置文件，淬火对应于一个触地现象。众所周知，淬火确实会发生。我们证明了{ \bf触地不能发生在介电常数分布的零点。}特别是，我们删除了在以前所有关于这个主题的工作中所做的关于触地集的紧性的假设，并且在大多数典型情况下，其有效性是未知的。这肯定地回答了{[ \it Y. Guo，Z. Pan和M. J. Ward，SIAM J. Appl. Math { \textbf{66}（2005）}，309- 338]}基于数值证据。结果关键取决于一个新的类型~I估计的淬火速率，我们建立。另外，在不作凸性假设的情况下，我们得到了达阵集紧的一些充分条件。这些结果在MEMS最优设计的应用中可能具有一定的定性重要性，特别是对于诸如微阀之类的器件。
This study is concerned with a family of parabolic system with general singular nonlinearities, which is a generalization of MEMS system. To some extent, the classification of global existence and quenching according to parameters and initial data is given. Moreover, the convergence rate is also obtained. We point out that compared to single MEMS equation, new ideas and techniques are introduced in obtaining the convergence rate for system in our study. In fact, due to the lack of variational characterization for the first eigenvalue of the linearized elliptic system, the methods in obtaining …
We consider a parabolic problem with Robin boundary condition which arises when the edge of a micro-electro-mechanical-system (MEMS) device is connected with a flexible nonideal support. Then via a rigorous analysis we investigate the structure of the solution set of the corresponding steady-state problem. We show that a critical value (the pull-in voltage) exists so that the system has exactly two stationary solutions when the applied voltage is lower than this critical value, one stationary solution for applying this critical voltage, and no stationary solution above the critical voltage.
Existence and nonexistence of solutions for fractional elliptic equations arising from closed MEMS model
Fudan University（复旦大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Jiangxi Normal University（江西师范大学）；Jiangsu University（江苏大学）
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. Initially we study the steady-state problem in one dimension and we present some numerical results regarding its bifurcation diagram. Next the $N-$dimensional nonlocal problem steady-state problem is investigated and a Pohozaev-type identity is first obtained which then facilitates derivation of an estimate of the pull-in voltage for radially symmetric domains. Later the time-dependent pr…
Existence and dynamic properties of a parabolic nonlocal MEMS equation
设$Ω\subset\mathbb{R}^n$为$C^2$有界域，$χ>0$为常数.我们将证明非局部MEMS方程$-Δv=\lam/(1-v)^2(1+χ\int_Ω1/(1-v)dx)^2$ in $Ω$，$v=0$ on $\1Ω$的常数$λ_N\geλ_N^{\ast}\geλ^{\ast}(1+χ\int_Ω\frac{dx}{1-w_{\ast}})^2$的存在性，使得对于任何$0\leλ<λ_N^{\ast}$都存在解，对于任何$λ>λ_N$都不存在解，其中$λ^{\ast}$是吸合电压，$w_{\ast}$是$Ω$中$-Δv=\lam/(1-v)^2$的最小解的极限，$v=0$ on $\1Ω$为$λ\nearrow λ^{\ast}$。我们将证明在$λ$上的各种有界性条件下，相应的抛物型非局部MEMS方程的整体解的存在性，唯一性和渐近行为。当$λ$较大时，我们还得到了抛物型非局部MEMS方程解的猝灭行为。
This paper is concerned with the elliptic equation $-Δu=\frac{λ}{(a-u)^p}$ in a connected, bounded $C^2$ domain $Ω$ of $\mathbb{R}^N$ subject to zero Dirichlet boundary conditions, where $λ>0$, $N\geq 1$, $p>0$ and $a:\barΩ\to[0,1]$ vanishes at the boundary with the rate ${\rm dist}(x,\partialΩ)^γ$ 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 qual…
