Global and touchdown behaviour of the generalized MEMS device equation

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 $\Omega \subset {\mathbb{R}}^n$ 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 $g^{\prime }(s)\le 0$ 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 $0\le \lambda <{\lambda }^{\ast }$ and has no solution for any $\lambda >{\lambda }^{\ast }$. 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 $0\le \lambda <{\lambda }^{\ast }$. 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 ${\lim }_{t\nearrow T}{\sup }_{\Omega }u(\cdot ,t)=1$.

25 pages

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