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 $u_t =Δu+λf(x)/g(u)$, $u<1$, in $Ω\times (0,\infty)$, $u(x,t)=0$ on $\partialΩ\times (0,\infty)$, $u(x,0)=u_0$ in $Ω$, where $Ω\subset\Bbb{R}^n$ is a bounded domain, $λ>0$ is a constant, $0\le f\in C^α(\overlineΩ)$, $f\not\equiv 0$, for some constant $0<α<1$, $0 0$ such that the associated stationary problem has a solution for any $0\leλ<λ^*$ and has no solution for any $λ>λ^*$. 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 $0\leλ<λ^*$. 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 $\lim_{t\nearrow T}\sup_Ωu(\cdot,t)=1$.