相关论文: Estimates for the quenching time of a parabolic eq…
The parabolic problem $u_t-\Delta u=\frac{\lambda f(x)}{(1-u)^2}+P$ on a bounded domain $\Omega$ of $R^n$ with Dirichlet boundary condition models the microelectromechanical systems(MEMS) device with an external pressure term. In this…
The singular parabolic problem $u_t-\triangle u=\lambda{\frac{1+\delta|\nabla u|^2}{(1-u)^2}}$ on a bounded domain $\Omega$ of $\mathbb{R}^n$ with Dirichlet boundary condition, models the Microelectromechanical systems (MEMS) device with…
We obtain upper bounds for the quenching time of the solutions of the nonlocal parabolic MEMS equation $u_t=\Delta u+\lam/(1-u)^2(1+\chi\int_{\Omega}1/(1-u) dx)^2$ in $\Omega\times (0,\infty)$, $u=0$ on $\1\Omega\times (0,\infty)$,…
We analyze the nonlinear elliptic problem $\Delta u=\frac{\lambda f(x)}{(1+u)^2}$ on a bounded domain $\Omega$ of $\R^N$ with Dirichlet boundary conditions. This equation models a simple electrostatic Micro-Electromechanical System (MEMS)…
We construct a quenching solution to the parabolic MEMS model \[ u_t = \Delta u - \frac{1}{u^2} \quad \text{in } \mathcal{B} \times (0,T), \quad u|_{\partial \mathcal{B}} = 1, \] where $\mathcal{B}$ is the unit disc in $\mathbb{R}^2$, and…
In this paper, we are interested in the mathematical model of MEMS devices which is presented by the following equation on $(0,T) \times \Omega:$ \begin{eqnarray*} \partial_t u = \Delta u +\displaystyle \frac{\lambda }{ (1-u)^2 \left( 1…
The objective of our paper is to investigate fractional elliptic equations of the form $(-\Delta)^s u=\frac{\lambda }{(a-u)^2}$ within a bounded domain $\Omega$, subject to zero Dirichlet boundary conditions. Here, $s\in(0,1)$, $\lambda>0$,…
A free boundary problem modeling a microelectromechanical system (MEMS) consisting of a fixed ground plate and a deformable top plate is considered, the plates being held at different electrostatic potentials. It couples a second order…
We consider a free boundary problem modeling electrostatic microelectromechanical systems. The model consists of a fourth-order damped wave equation for the elastic plate displacement which is coupled to an elliptic equation for the…
This work investigates a mathematical model arising in the study of MEMS devices, described by the following parabolic equation on $[0,T)\times\Omega$: $$\partial_t v = \Delta v + \frac{\lambda}{(1-v)^2\left( 1 + \gamma \int_{\Omega}…
The occurrence of a finite time singularity is shown for a free boundary problem modeling microelectromechanical systems (MEMS) when the applied voltage exceeds some value. The model involves a singular nonlocal reaction term and a…
This paper is concerned with the nonlinear elliptic problem $-\Delta u=\frac{\lambda }{(a-u)^2}$ on a bounded domain $\Omega$ of $\mathbb{R}^N$ with Dirichlet boundary conditions. This problem arises from Micro-Electromechanical Systems…
We consider a nonlocal parabolic model for a micro-electro-mechanical system. Specifically, for a radially symmetric problem with monotonic initial data, it is shown that the solution quenches, so that touchdown occurs in the device, in a…
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 dynamical and stationary behaviors of a fourth-order equation in the unit ball with clamped boundary conditions and a singular reaction term are investigated. The equation arises in the modeling of microelectromechanical systems (MEMS)…
We study the effect of the parameter $\lambda$, the dimension $N$, the profile $f$ and the geometry of the domain $\Omega \subset\mathbb{R}^N$, on the question of uniqueness of the solutions to the following elliptic boundary value problem…
The evolution problem for a membrane based model of an electrostatically actuated microelectromechanical system (MEMS) is studied. The model describes the dynamics of the membrane displacement and the electric potential. The latter is a…
We analyze nonnegative solutions of the nonlinear elliptic problem $\Delta u=\frac{\lambda f(x)}{u^2}+P$, where $\lambda>0$ and $P\geq0$, on a bounded domain $\Omega$ of $\mathbb{R}^N$ ($N\geq 1$) with a Dirichlet boundary condition. This…
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,…
Well-posedness of a free boundary problem for electrostatic microelectromechanical systems (MEMS) is investigated when nonlinear bending effects are taken into account. The model describes the evolution of the deflection of an electrically…