The singular parabolic problem $u_t=Δu -\frac{λf(x)}{(1+u)^2}$ on a bounded domain $Ω$ of $R^N$ with Dirichlet boundary conditions, models the dynamic deflection of an elastic membrane in a simple electrostatic Micro-Electromechanical System (MEMS) device. In this paper, we analyze and estimate the quenching time of the elastic membrane in terms of the applied voltage --represented here by $λ$. As a byproduct, we prove that for sufficiently large $λ$, finite-time quenching must occur near the maximum point of the varying dielectric permittivity profile $f(x)$.
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…
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 pro…
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。
A model for a MEMS device, consisting of a fixed bottom plate and an elastic plate, is studied. It was derived in a previous work as a reinforced limit when the thickness of the insulating layer covering the bottom plate tends to zero. This asymptotic model inherits the dielectric properties of the insulating layer. It involves the electrostatic potential in the device and the deformation of the elastic plate defining the geometry of the device. The electrostatic potential is given by an elliptic equation with mixed boundary conditions in the possibly non-Lipschitz region between the two plate…
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 …
A stationary free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a nonlocal fourth-order equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the free domain between the two plates. The strength of the coupling is tuned by a parameter $λ$ which is proportional to the square of the applied voltage difference. Existence of a stable stationary s…
A parabolic free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a fourth-order semilinear parabolic equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the device. The strength of the coupling is tuned by a parameter $λ$ which is proportional to the square of the applied voltage difference. It is proven that the model is locally well-posed i…
We analyze the nonlinear elliptic problem $Δu=\frac{λf(x)}{(1+u)^2}$ on a bounded domain $Ω$ of $\R^N$ with Dirichlet boundary conditions. This equation models a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at -1. When a voltage --represented here by $λ$-- is applied, the membrane deflects towards the ground plate and a snap-through may occur when it exceeds a certain critical value $λ^*$ (pull-in voltage). This creates a so-called "pull-in instability" which gr…
In this paper, we are interested in the mathematical model of MEMS devices which is presented by the following equation on $(0,T) \times Ω:$ \begin{eqnarray*} \partial_t u = Δu +\displaystyle \frac{λ}{ (1-u)^2 \left( 1 +\displaystyle γ\int_Ω \frac{1}{1-u} dx \right)^2}, \quad 0 \leq u <1, \end{eqnarray*} where $Ω$ is a bounded domain in $\mathbb{R}^n$ and $λ, γ> 0$. In this work, we have succeeded to construct a solution which quenches in finite time T only at one interior point $a \in Ω$. In particular, we give a description of the quenching behavior according to the following final profile $…
Existence and dynamic properties of a parabolic nonlocal MEMS equation
设$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。
We study a nonlinear wave equation appearing as a model for a membrane (without viscous effects) under the presence of an electrostatic potential with strength $λ$. The membrane has a unique stable branch of steady states $u_λ$ for $λ\in\lbrack0,λ_{\ast}]$. We prove that the branch $u_λ$ has an infinite number of branches of periodic solutions (free vibrations) bifurcating when the parameter $λ$ is varied. Furthermore, using a functional setting, we compute numerically the branch $u_λ$ and their branches of periodic solutions. This approach is useful to validate rigorously the steady states $u…
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…
We study general equations modeling electrostatic MEMS devices \begin{equation} \begin{cases} \label{P} φ\big(r,- u'(r)\big)=λ\int_0^r\frac{f(s)}{g(u(s))}\,\mathrm{d}s, & r\in(0,1), \\ 0 < u(r) < 1, & r\in(0,1), \\ u(1) = 0, \tag{$P_λ$} \end{cases} \end{equation} where $φ$, $g$, $f$ are some functions on $[0,1]$ and $λ>0$ is a parameter. We obtain results on the existence and regularity of a touchdown solution to \eqref{P} and find upper and lower bounds on the respective pull-in voltage. In the particular case, when $φ(r,v) = r^α|v|^βv$, i.e., when the associated differential equation involve…
The abstract quasilinear Cauchy problem for a MEMS model with two free boundaries
在本文中，我们重新制定了一个理想的静电驱动的MEMS器件的动力学与两个弹性膜作为一个抽象的准线性发展方程的初值问题的数学模型。应用压缩映射定理，它表明，该模型是局部适定的时间为任何值的设备的源电压。此外，它被证明，这里考虑的MEMS模型具有全局解的小电源电压，而大电源电压的模型的解决方案有一个有限的最大存在时间。此外，我们评论的关系，我们的模型，它的固定版本和它的小纵横比限制，存在一个独特的指数稳定的稳定状态，并证明收敛到一个解决方案的窄间隙模型消失的纵横比限制。我们的结果扩展了[Martin Kohlmann：On an elliptic-parabolic MEMS model with two自由边界，2014]中提出的椭圆抛物MEMS模型的讨论，导致半线性抽象发展方程的柯西问题。
