Global and touchdown behaviour of the generalized MEMS device equation

TL;DR

This paper establishes the existence, non-existence, and long-term behavior of solutions to a generalized MEMS equation, including conditions for touchdown phenomena and convergence to stationary states.

Contribution

It proves local and global existence, identifies a critical parameter for solution existence, and analyzes touchdown behavior for the generalized MEMS model.

Findings

Existence of a critical parameter mbda^* for stationary solutions.

Global solutions converge to stationary states as time approaches infinity.

Conditions under which solutions experience touchdown at finite time.

Abstract

We prove the local and global existence of solutions of the generalized micro-electromechanical system (MEMS) equation $u_t =\Delta u+\lambda f(x)/g(u)$, $u<1$, in $\Omega\times (0,\infty)$, $u(x,t)=0$ on $\partial\Omega\times (0,\infty)$, $u(x,0)=u_0$ in $\Omega$, where $\Omega\subset\Bbb{R}^n$ is a bounded domain, $\lambda >0$ is a constant, $0\le f\in C^{\alpha}(\overline{\Omega})$, $f\not\equiv 0$, for some constant $0<\alpha<1$, $0<g\in C^2((-\infty,1))$ such that $g'(s)\le 0$ for any $s<1$ and $u_0\in L^1(\Omega)$ with $u_0\le a<1$ for some constant $a$. We prove that there exists a constant $\lambda^{\ast}=\lambda^{\ast}(\Omega, f,g)>0$ such that the associated stationary problem has a solution for any $0\le\lambda<\lambda^*$ and has no solution for any $\lambda>\lambda^*$. We obtain comparison theorems for the generalized MEMS equation. Under a mild assumption on the initial value we prove the convergence of global solutions to the solution of the corresponding stationary elliptic equation as $t\to\infty$ for any $0\le\lambda<\lambda^*$. We also obtain various conditions for the existence of a touchdown time $T>0$ for the solution $u$. That is a time $T>0$ such that $\lim_{t\nearrow T}\sup_{\Omega}u(\cdot,t)=1$.