Bulletin of the
Korean Mathematical Society BKMS

In this paper, we study nonlinear equation arising in MEMS modeling electrostatic actuation. We will prove the local and global existence of solutions of the generalized parabolic MEMS equation. We present that there exists a constant $\lambda^{*}$ such that the associated stationary problem has a solution for any $\lambda < \lambda^{*}$ and no solution for any $\lambda > \lambda^{*}$. We show that when $\lambda < \lambda^{*}$ the global solution converges to its unique maximal steady-state as $t\rightarrow\infty$. We also obtain the condition for the existence of a touchdown time $T\leq\infty$ for the dynamical solution. Furthermore, there exists $p_0>1$, as a function of $p$, the pull-in voltage $\lambda^{*}(p)$ is strictly decreasing with respect to $1p_0$.

Keywords: MEMS equation, upper-and-lower solution method, global convergence, touchdown time

MSC numbers: 35K55, 35K65, 35B40