Global and touchdown behaviour of the generalized MEMS device equation

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 $Omegatimes (0,infty)$, $u(x,t)=0$ on $partialOmegatimes (0,infty)$, $u(x,0)=u_0$ in $Omega$, where $OmegasubsetBbb{R}^n$ is a bounded domain, $lambda >0$ is a constant, $0le fin C^{alpha}(overline{Omega})$, $f otequiv 0$, for some constant $0<alpha<1$, $0<gin C^2((-infty,1))$ such that $g(s)le 0$ for any $s<1$ and $u_0in L^1(Omega)$ with $u_0le a<1$ 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 $0lelambda<lambda^*$ 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 $ttoinfty$ for any $0lelambda<lambda^*$. 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 $lim_{t earrow T}sup_{Omega}u(cdot,t)=1$.