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 $\Omega\subset\Bbb{R}^n$ 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 $\lim_{t\nearrow T}\sup_{\Omega}u(\cdot,t)=1$.